isotropic manifolds of indefinite metric

Isotropic Manifolds o f Indefinite Metric 1)

by Jos~.P~ A. WOLF (Berkeley, California)

Chapter I.

Chapter I I .

Contents Page

23 w 0. In t roduct ion . . . . . . . . . . . . . . . . . . w 1. Preliminaries on pseudo-R:EMA~Nian manifolds, LrE

groups and curvature . . . . . . . . . . . . . . 24

Locally and globally isotropic pseudo-R:EMA~ian manifolds 30

w 2. The model spaces . . . . . . . . . . . . . . . . 30 w 3. Groups transit ive on cones and quadrics . . . . . . 39 w 4. Structure of locally isotropic manifolds . . . . . . . 43 w 5. The universal covering theorem . . . . . . . . . . 47 w 6. A part ial classification of complete locally isotropic

manifolds . . . . . . . . . . . . . . . . . . . . 47 w 7. The classification theorem for isotropic manifolds . 50

Homogeneous locally isotropic pseudo-RiEMAN~ian manifolds . . . . . . . . . . . . . . . . . . . . . . 52

Application of the classification in constant curvature 52 The classification theorem for complete homogeneous locally isotropic manifolds of noncons tant curvature 55 A characterization and completeness criterion for locally symmetr ic affine homogeneous spaces . . . . . 57 A tool for the construction of incomplete manifolds . 59 Existence of incomplete homogeneous spaces . . . . 64

w 8. w 9.

w lo.

w w 12.

2) This work was begun at the Summer I n s t i t u t e o n Relativity and Differential Geometry, Santa Barbara, 1962. I t was partially supported by National Science Foundation Grant GP-812.

22 JOSEPH A. WOLF

O. Introduction

In an earlier paper [17] we studied the pseudo-RIEMAN~ian manifolds of constant sectional curvature and classified the complete homogeneous ones except in the fiat case; this paper is a more thorough s tudy of a larger class of manifolds.

A pseudo-RI~,MAN~ian (indefinite metric) manifold is (locally) isotropic if, given two nonzero tangentvectors of the same "length" at the same point, there is a (local) isomctry carrying one vector to the other. The local version of this property is automatic in the case of constant curvature. Chapter I begins with the study of certain isotropic spaces, called model spaces, which are fiat or are indefinite metric analogs of the real, complex and quaternionic elliptic and hyperbolic spaces and of the CAYLv.u elliptic and hyperbolic planes. In w 4 it is shown that every locally isotropic manifold is locally isometric to a model space; an interesting consequence, extending a result of S.HELOASOI~ ([8], w 4) in the LORENTZ case, is that the manifold must be of constant curvature if either the number of negative squares in the metric, or the number of positive squares in the metric, is odd. This local isometry extends to a metric covering in the case of complete manifolds (w 5); as consequences we obtain a global classification of the complete locally isotropic manifolds of nonconstant curvature in about half the signatures of metric (w 6), and we obtain a global classification of the isotropic manifolds (w 7).

Chapter I I is devoted to homogeneous locally isotropic manifolds. Using the techniques of [17] (w 8), the complete ones are classified (w 9). In w 12 it is shown that every isotropic p s e u d o - l ~ I ~ . ~ c i a n manifold of strictly indefinite metric contains an (necessarily locally isotropic) open submanifold which is homogeneous but not complete. This is surprising in view of the RIEMA~ian case where homogeneity implies completeness. I t shows that one must add the hypothesis of completeness to Theorem 12 of [17] and Theorems 1 and 2 of [19].

I am indebted to Professor JAcques TITS for a conversation on the indefinite CAYLWY plane. Professor TITS informs me that he has an unpublished classification of the isotropic pseudo-RI~MA~cNian manifolds obtained by means of a classification of subalgebras ~ of small codimension in a simple LIE algebra ffi.

Note. I t is possible to omit reading w167 1 . 3 . 3 - 1 . 3 . 5 , w 2.9, w167 3 . 3 - 3.4 and w 10 while reading the rest of this paper. The main results are stated at the beginnings of w167 4, 5, 6, 7, 9 and 12.

Isotropic manifolds of indefinite metric

1. Preliminaries on pseudo-RIEMANNian manifolds, LIE groups

and curvature

23

1.1. Pscud0-RIEMANNian manifolds. A pseff.Zt, o-RIEMA~WVia~ ~ n i f o ~ is a pair M~ ~ (M, Q) where M is an n-dimensional differentiable manifold, Q is a differentiable field of real nonsingular symmetric bilinear forms Q. on the

h n

tangentspaces M , of M, and each Q. has signature - 27 x~ ~- 27 x~. We often 1 h+l

say ~Y/~ when 21I is intended, and then the i~se~o-RiBM~c~ian metric Q is understood. Given X �9 M~, we usually write I IX I] ~ for Qx(X, X). The LBvz-ClVlT, a connection is the unique torsionfree linear connection on the tangentbundle of M~ such that parallel translation is a linear isometry (preserves the inner products Q,) on tangentspaces; M~ is complete if this connection is complete, i.e., if every geodesic can be extended to arbitrary values of the affine parameter.

An isometry of pseudo-RIEMA~ian manifolds is a diffeomorphism f : Mh"--~N' ~ which induces linear isometrics on tangentspaces. The collection of all isometrics of M~ (onto itself) forms a LIE group I(M~) called the full group of isometrics of M~; its identity component Io(M~) is the connected group of isometrics. Given x e M~, we consider the collection of all local maps which are each an isometry (in induced pseudo-RIEM),~TNian structure) of an open neighborhood of x onto another open neighborhood and which leave x fixed, where two of these local maps are identified if they agree on a neighborhood of x; this collection of germs of maps forms a LIE group which is faithfully represented on the tangentspace (M~)~, and is called the group of local isometries at x; its LIE algebra is the subalgebra consisting of elements which vanish at x in the LIE algebra of germs of KILLINO vectorfields at x.

M~ is homogeneous ifI(M~) is transitive on its point.s; M~ is locally homogeneous if, given x and y in M~, there is an isometry of a neighborhood of x onto a neighborhood of y which carries x to y. M~ is symmetric (resp. locally symmetric) if, given x ~ M~, there is an s, ~ I(M~) (resp. local isometry s, at x) of order 2 and with x as isolated fixed point; s , is then the symmetry (resp. local symmetry} at x. M~ is isotropic (resp. locally isotropic) if, given x ~ M~ and nonzero tangentvectors X, Y at x with IIX]I * --~ II Yll 2, there exists g eI(M~) leaving x fixed (resp. a local isometry g at x) such that g . X = Y.

I f M~ is connected and either isotropic or symmetric, then it is both homogeneous and complete. For homogeneity one connects any two points x and y by a broken geodesic arc, chooses isometrics fixing the midpoints (in affine parameter) and reversing the tangentvector there for each of the geodesic segments, and notes that the successive product of these isometrics sends x

24 JOSEPH A, WOLF

to y. For completeness, one extends a geodesic via its image under any iso- me t ry which fixes a point of the geodesic and reverses the tangentvector there. Combining these two arguments and applying the H~crN~-Bo~v.L Theorem, one can also see: I f M~ is connected and is either locally isotropic or locally symmetric, then M~ is locally homogeneous. In fact we will soon see t ha t M~ must be (locally) symmetr ic ff it is (locally) isotropic.

A p s e u d o - R ~ M A ~ i a n covering is a covering g : N~--* M~ of connected pseudo-RIE~ANNian manifolds where ~ is locally an isometry. Then each deck t ransformat ion (homeomorphism d : N~ --~ N~ such t h a t ~ . d ~ ~) is an i sometry of N~. I f one of N~ and M~ is locally homogeneous (resp. locally symmetr ic , resp. locally isotropic, resp. complete) then x carries the same proper ty over to the other. I f M~ is homogeneous (resp. symmetr ic , resp. isotropic), then we can lift each isometry and N~ has the same property. Bu t these global properties usual ly do not descend.

1. 2. Reductive groups. We assume famil iar i ty with LI~ groups. I f G is a LIE group, then G O denotes the iden t i ty component, ffi denotes the LIE algebra, ad and Ad denote the adjoint representations of Go and (b on ffi, and exp is the exponential map (~ --> G o . The following conditions are equivalent, and G and ffi are called reductive if one of them is satisfied: (1) ad (Go) is fully reducible on ffi, (2) Ad(ffi) is ful ly reducible on ffi, (3) ffi z ffi' �9 9/ where 9 / i s the center and ffi' is a semisimple ideal (the semisimple part), (4) ~ has a faithful ful ly reducible linear representation, (5) ffi has a nondegenerate real symmetr ic ad(G0)-invariant bilinear form which is the trace form of a faithful linear representation. I f H is a closed subgroup of G, it is called reductive in G i f ada(H) is fully reducible on ffi; .~ is reductive in ffi if H 0 is reductive in G; an element g E G is semisimple if ad (g) is fully reducible.

The following well-known fact is immediate from SCHUR'S L e m m a because G is reductive; we will use it often.

1 . 2 . 1 . Lemma. Let G be an irreducible closed connected group o[ linear trans/ormations o/ a real vectorspaee V, let Z be the center o/ G, and let G ~ be the derived group o/ G. Then (a) G is semisimple, or (b) G ~-- G' . Z where Z is a circle group which gives a complex structure to V , or (c) G ~ G'- Z where either Z consists o /a l l nonzero real scalar trans/ormations, or Z is generated by a circle group and the nonzero real scalars; case (c) does not occur i/ V has a nondegenerate G-invariant bilinear /orm.

A C~R~'~ involution of a semisimple LIE algebra ffi is an involutive automorphism of ffi whose fixed point set is the LIE algebra of a maximal compact subgroup of the adjoint group of ffi; any two are conjugate by inner automorphisms of ~ . We will sometimes say CXRT•• involution for an involutive

Isotropie manifolds of indefinite metric 25

automorphism of a reduetive LIE algebra which induces a C~TA~r involution on the semisimple part. I f .~ is an algebraic reductive subalgebra of a finear semisimple LIE algebra ~5, then it follows (see [4]) from G.D.MosTow's treat- ment of the case of semisimple .~ [14] that there is a CARTA~ involution of ~5 which preserves ~ and induces a CARTAN involution of .~.

1. 3. Curvature. Let M~ be a pseudo-RIEMAN~ian manifold. I f S is a 2-dimensional subspace of a tangentspace (M~)~ on which Q~ is nondegenerate (nonsingular subspace), and if ~-~ is the curvature tensor of the LEvI-CIVITA connection, and is viewed as a transformation, then the sectional curvature of M~ along S is given by

Q~(.~(X, Y) X, Y) K(S) --~ -- Q~(x, x ) .Q~(Y, Y) - Q~(x,Y) 2 "

M~ has constant curvature ]c (at x) if K(S) ~ ]c for every nonsingular tangent 2-plane S to M~ (at x). I f the group of local isometries at x contains the identity component SO h (n) of the orthogonal group of Q~, then M~ is easily seen to have constant curvature at x. This is particularly useful when M~ is connected and n ~ 3, for then F. ScnvR's Theorem says that M~ has constant curvature if it has constant curvature at every point. M~ is called/lat if it has constant curvature zero. Finally, M~ is locally symmetric if K (S) does not change as S undergoes parallel translation, i.e., if U-2 is parallel.

Part 2 of the following result generalizes the theorem on sign of curvature of a RIEMANNian symmetric space.

1.3.1. Theorem. Let M ~- G/H where H is a closed subgroup o/a semisimple LIE group G such that .~ is invariant under a C~RTa~ involution a o/ ~ . Let B be an ad (G)-invariant a-invariant nondegenerate symmetric bilinear ]orm on (~; B ~-- Za iB i where the B i are the KXZLXNa [orms o/ the simple ideals o/ (~ ; suppose that each a~ ~ O. Then B is nondegenerate on ~ and:

1. Let ~ :- ~ ( • relative to B). Then (~ ~- ~ ~- ?0~ is an ad(H)-stable a-stable vectorspace direct sum, ?0~ is naturally identi]ied with a tangentspace o/ M, and B induces a nondegenerate ad (H)-invariant bilinear /orm on ~l~ which de/ines a G-invariant pseudo-RzEMa~ian metric Q on M .

2. Let S be a a-invariant 2-dimensional linear subspace o] ~0~. Then S is nonsingular /or B, so the sectional curvature K (S) o~ (M , Q) along S is de/ined, and K (S) ~ O.

3. The sectional curvatures o/ (M, Q) are bounded/tom above and below, i] and only i/ (M, Q) has every sectional curvature non-positive.

4. I / 9~ has a nonsingular 2-dimensional subspace S with orthonormal basis {X, Y) andanelement Z I S with [ X , Z ] -~ O and II YI] ~" IIZI] 2 < 0, then (M, Q) has a positive sectional curvature.

26 JOSEPH A. WOLF

Remark 1. I f we had stipulated a~ < 0, then all signs of curvature would be reversed. I f M is symmetric as a coset space, so ~ is the fixed point set of an involutive automorpbSsm w of {B, then .~ is reductive because ~5 and w are semisimple, and ~ is algebraic as defined by w, so it is automatic that ~ is invariant under a CART~_N involution of (~.

Remark 2. Parts 2 and 4 lead us to Delinition. In the notation of Theorem 1.3.1 a 2-plane in ~ is involutive

if it is invariant under a CARTAI~ involution of ~ which preserves ~ , and the involutive sectional curvatures of (M, Q) are the sectional curvatures along involutive 2-planes.

Now the sign of involutive sectional curvature is constant for many manifolds, but it is unusual for the sign of sectional curvature to be constant. In fact, from Theorems 2.9 and 4.1 it will follow that a connected locally isotropic pseudo-R:~MA~ian manifold M~ has sectional curvatures of one sign, if and only if either h : 0 or h : n (definite metric), or M~ is of constant sectional curvature.

1 .3 .2 . Proof. f f~:ff~+~-(~_ where a is q-1 on ff~+ and --1 on ~_ and where B is negative definite on ff~+ and positive definite on ~5_. a(~)--- implies ,~ = (~ n (~+) -k (~ • ff~-), so B is nondegenerate on ~ . Now the first statement is obvious. According to NoMIzu [15] the ad(G)-invariance of B implies that the curvature tensor :~ of (M, Q) is given by

(X, Y)Z = �88 {IX, j r , Z ] ~ ] ~ - [Y, [X,Z]~]~ - 2 [IX, Y]~,Z]~ - 4 [IX, Y]a, Z]~}

where X , Y, Z E ~J~ and subscripts denote B-orthogonal projections. For X ----- Z the second term drops out and the first is minus half the third. I f X and Y span a nonsingular plane S and if B (X, Y) = 0, then it follows that IlXll ~ r o r Yll ~ and that

l l X l r . II YII ~" K(S) -~ - B(~-~(X, Y )X , Y) : �88 B([[X, Y]~, X]~ , Y) - l -B( [ [X , Y]$, X]~ , Y)

: �88 B([[X, Y]m, X], Y) ~- B([[X, Y]~, X], Y)

---- �88 B([X, Y]m, [X, Y]) + B([X, Y]~, IX, Y])

: �88 B([X, Y]~, [X, Y]m) --k B([X, Y]5, [X, Y]~).

Now let Z ----- [X, Y]. We have just seen that

llXll~. II Y[[~. K(S) : �88 B(Zm, Z~) q-- B(Z~, Z~) .

Suppose further that ~(S) = S. We may then choose the basis {X, Y} of S such that a ( X ) = o ~ X and a( Y) ----- fl Y where ~----- =t= 1 and f l = =k 1; in addition we may assume [[X][ 2 and [[ y[[2 to have absolute value 1, and we

Isotropic manifolds of indefinite metric 27

may continue to assume B ( X , Y ) = O . Then IIXII 2-- -c~, IIYII ~-- - f l and a ( Z ) = o ~ f l Z . I f ~fl--~ 1, then ] I Z ~ I I 3 < 0 and IIZ~II3<0, and it follows that K ( S ) < O . I f c~fl-----1 then I I Z ~ l l 3 > 0 and IIZ~II2:>0, and it follows that K(S) __< 0. The second statement is now proved.

1 .3 .3 . Suppose that the sectional curvatures of (M, Q) are bounded from above and below, and let S be a nonsingular 2-plane in ~9~. We must prove K (S) <_ O.

Given Z ~J~, we write Z : Z+-}-Z_ where a ( Z + ) : Z+ and a(Z_)----- ---- - Z_, and we write Z t ---- Z+ q- tZ_ for every real number t. Suppose that S has a basis {X, Y} such that X_L Y~ for every t and {X, Y~} spans a 2-plane S t. We then define

D,-~ B (X , X ) . B (Yt , Y~) - B (X , y~)3 ~_ ]]X]]3(]] Y+]r -}- t21] Y-]I 3) =Do-{ - tzb

N, = �88 II [ x , r , ] ~ II ~ + II [ z , r , ]~ 113 = No + t: (�88 II [X, r _ l ~ l l 3 + IlEX, r_]~ll 3)

q- t(terms which vanish if a(X) : -F X ) ---- No q- t2u - F t v and have K(S+) : Nt /D ~ if Dt :/: 0.

I f a ( S ) : S, we have seen that K(S) < 0. Now suppose a(S)5~ S. I f a(S) ,~ S :/: 0, then we have 0 @ X ~ S with a(X) : 4- X. Suppose first that a ( X ) : X . Then S has basis {X,Y} with X_L Y+ and Y t : / : 0 @ Y - , and e a c h S t i s a P l a n e . D + : D o-Ft~b with D o > 0 and b < 0 , so D~ has precisely two zeros, D a : D _ ~ : 0 , a > 0. N ~ : N 0 q - t 2u with N o < 0 because K (So) < 0, and with u > 0. N+ must vanish if D r : 0 by boundedness of K(S,) for s near t. I f N+ is identically zero, then K(S) : 0. Other- wise N~ changes sign with D~ and K ( S ) < 0. A similar argument holds if ~(X) = - X .

Now suppose a(S) ,~ S = 0, and choose a basis {X, Y} of S with II X[13:/=0 and X • Y+. Suppose first that ]1X I13 > 0, and let T be the span of X and I/_. We do not generally have X _l_ Yt, so D~ is of the form Dt = Do -[-

ta ~ f2b. D~ has two zeros for So is positive definite, and, for It I large, S t is indefinite because II Y, II ~ < 0. We may assume N~ not identically zero; then Nt -~ N o q- tv -}- t3u has at most two zeros, and boundedness of the defined K(S~) shows that a zero of D~ is a zero of N t. Now we have D~ =

r ( t - - x ) ( t - - y ) and N~-----s(t- x ) ( t - y ) because the zeros x and y of Dt are distinct, and K(S~) = N~/D~ = s/r when D~ r 0. As K(S0) < 0 by the last paragraph, it follows that K ( S ) ~_ O. The necessity in the third statement is proved.

1 .3 .4 . Suppose that (M, Q) has every sectional curvature non-positive; we must prove the sectional curvatures bounded. I f the sectional curvatures

28 JOSEPH A~ WOLF

of (M, Q) are not bounded, then there is a sequence {P~} in the GRASSMANN manifold L of nonsingular 2-planes of some signature in ~R, such that {K (P~)}-~ -oo . We may assume {P~}-+ S for some necessarily singular 2-plane S in ~[R because the GRASSMA~-S manifold of all 2-planes in ~ is compact.

Suppose first that S is not totally isotropic. Then S has a basis {X, Y} such tha t either {X+, Y+} or (X_, Y_} is an orthonormal basis of a plane, and I IX l [~a 0. We may assume (X+, Y+} orthonormal, and we have S,, D , : D o § 2 4 7 with D O r and N t ~ N o § 2 4 7 as in w with K ( S ~ ) ~ N~/D~ whenever D~ r 0. D~ changes sign at t -~ 1, and each P~ has an element Z i with II Zi[I ~ ~ It X[[~; this gives a sequence {t~}-+ 1 such that (Si ~ St~ ~ L) {K(S~)}-+ --oo, by choice of S. In particular N~ is not constant. Our hypothesis on sign of curvature implies that N t and D, change sign simultaneously. I f D t is linear, it follows that N~ is linear and both are constant multiples of t -- 1 ; then K(S~) is constant, which is impossible. I f D t is not linear, then its zeros are distinct because it changes sign at t ---- 1, and again K(S~) is constant. This is a contradiction.

Now suppose S totally isotropic; then S has basis (X, Y} such that both {X+, Y+} and {X_, Y_} are orthonormal. Let Q,,, be the plane spanned by X, and Yr.

n, . , ~- B ( X , , X , ) . B( Y,, Y,) - B ( X , , y,)2 __ (s~ _ 1) (t ~ - 1)

and the corresponding N,,~ is quadratic in s and quadratic in t. For fixed t ~: -~- 1, Ds, t changes sign as s crosses :k 1 ; thus the same is true of N,., and so s 2 - 1 divides N,,~. Similarly t ~ - 1 divides N,,, , and it follows that K(Q,,t) is constant for Is[ :~ 1 @It [ . That is impossible because L contains a sequence of Qs,, converging to S. As before, this contradicts con- stancy of the K(Q,,~).

The third statement is proven.

1 . 3 . 5 . Let X , Y, Z E 9J~ be mutually or~hogonal such that plane S spanned by X and Y has K(S) @ O, IX, Z] = 0 and II YII z" I lZl l ~ < 0. Then the plane T, spanned by X and Y § tZ is nonsingular for large t, and the formula for curvature then gives

IlXl] ~" II YII ~" K(S) = }lXl} ~, II Y § tZtl ~" K(TO

because ( Y § and [X, Y § Y]. For large t,

I1 Y § tZl[ 2" lI YII ~ < 0 ,

so K (T,) has sign opposite that of K (S). Thus one of them must be positive. This completes the proof of Theorem 1.3.1. Q . E . D .

Isotropic manifolds of indefinite metric

Chapter I

Locally and Globally Isotropie Pseudo-RIEMANNian Manifolds

29

This Chapter develops the basic theory of isotropic pseudo-RIE~NNian manifolds. Important examples are described in w 2. In w 4 the local structure of locally isotropic manifolds is described in terms of these examples, and w 5 shows that these examples act as metric covering spaces for complete locally isotropic manifolds. w 6 extends that covering theorem to a partial classification, and w 7 gives the global classification of isotropic pseudo-RIEMXNman manifolds.

2. The model spaces

We shall describe spaces analogous to the isotropic RIEMAN~ian manifolds. Our later analyses of isotropic pseudo-RIEMA~ian manifolds will depend on comparison with the spaces described below. The spaces R~ described below are fiat complete connected simply connected isotropic pseudo-RIEMANNian manifolds in an obvious way ([17], w 3).

2.1. Indefinite unitary groups over division algebras. Let F be a real division algebra R (real), C (complex) or K (quaternion). There is a natural conjugation x--~ x of F over R; this defines the multiplicative group F' of unmodular (xx = 1) elements of F, and defines the notion of hermitian form on an F- vectorspace. Given integers 0 < h < n, F~ will denote the right F-vectorspace of n-tuples together with the hermitian form

b ( ( x i . . . . . Xn), (Yl . . . . . Yn)) : g~' Xi~]i"

The "indefinite uni tary" group over F, U a (n, F), is defined to be the group of automorphisms of F~. The usual notation for indefinite unitary groups over F is:

U a(n, R) ~ 0 n (n), indefinite orthogonal group

U a(n, C) = U a(n), indefinite unitary group

US(n, K) = Spa(n), indefinite symplectic group

If h = 0 or h ~- n, one omits writing it. Finally, SO n (n) is defined to be the identity component of 0 n(n), and SU n (n) consists of the elements of U a(n) of determinant 1.

2.2. Indefinite GRASSMANN manifolds over division algebras. I f h' ~ h and n' - h' _~ n - h then there is an inclusion F~: c F~ consistent with the

30 JOSEPH A. WoLF

hermitian forms; it is unique up to action of an element of Uh(n, F), i.e., U h(n, F) acts transitively on the collection of all subspaces F~: of F~. The subgroup of Uh(n, F) leaving invariant a particular F~: is isomorphic to U h-h' (n - n ' , F) • U h' (n', F). Thus we may view the collection of all F~; in F~ as the coset space

flh,,h;,,,,(F) : U~(n, F)/{Uh-h'(n - n', F) • Uh'(n ', F)}. (2.2.1)

Gh, h;.,,.(F) is the quotient of a L:~ group by a closed subgroup, and thus carries the structure of a differentiable manifold; it has dimension /(n -- n ' )n ' where ] is the dimension of F over R.

For the moment we write (2.2.1) above as M = G / H , where H is the isotropy group at p ~ M. Then H has an element s, defined to be the identity I on p and -- I on p i , such that s 2 ---- I (on F~) and H is the full centralizer of s in G. In other words, M is a symmetric coset space and LIE algebras satisfy ffi ---- ~ § ~ where .~ is the eigenspace of ~ 1 for ad (8) and ~ is the eigenspace of -- 1 ; .~ ---- ~J- here with respect to the KILLT~G form B of ffi, and B is nondegenerate on ~ . The restriction of - B to ~ thus endows M with a G-invariant pseudo-RIEMA~ian structure. This structure has signature (a,b) where n " --- n - - n ' , h " = h - - h ' , a : / ( h ' ( n '~ - h ' ) + h" ( n ' - h ' ) )

and b = / ( h ' h " + ( n ' - h ' ) ( n " - h" ) ) : ] ( n - n ' ) n ' ; it has non-negative involutive sectional curvature. I f we use B, rather than - B, to endow M with a metric, then the roles of a and b are interchanged and we obtain a pseudo-RIg~A~-tdan structure of non-positive involutive sectional curvature. M is always viewed with one of these structures, and is called an indefinite GRASS~IA~N manifold over F.

2. 3. The indefinite elliptic space P~ (F) is defined to be the GRASS~A~I~ manifold of all F01 in F~ +i with the metric of non-negative involutive sectional curvature; it has d imens ion/n and signature ( / h , / ( n - h)) ; if h : 0 it is the ordinary elliptic space over F, and if h = n it is the usual hyperbolic space, with metric reversed, over F. An obvious modification of the coset representation (2.2.1) yields

P~(R) : S0a(n ~- 1)~Oh(n) (2 .3 .1)

P~(C) = SUh(n + 1)/Ua(n) (2.3.2)

P~ (K) ---- Sp h (n % 1)/(Sp h (n) x Sp (1)} (2 .3 .3)

In the representation P~ (F) ----- G / H immediately above, we still have the decomposition ~ = .~ ~ ~ , and the negative of the K:LLrNG form of ffi gives ~) the structure of F~. The transformations of ~ generated by ad(H)


are easily seen to be precisely the group generated by U h (n, F) together with the scalar multiplications by elements of F'. In particular, P~(F) is an isotropic pseudo-RI~MA~Nian manifold M ~ , the pseudo-RIEMA~-Nian analog of the elliptic space 1 ~ (F) .

P~(F) is complete because it is globally symmetric. I f F 9a R, it is simply connected, as is seen by its representation (2.3.2) or (2.3.3) as coset space of a simply connected group by a closed connected subgroup. P~(R) is the manifold S~/{ =t= I} of [17] with universal pseudo-Rx~MA~Nian covering manifold S~ described in ([17], w 4.4 and w 11.2).

2. 4. The indefinite hyperbolic space H~(F) is the GRASSMA~-N manifold of all F10 in ~-hvn+l with the metric of non-positive involutive sectional curvature; one obtains it by replacing the metric with its negative on P~_h(F). Thus we have

H~(R) ~-- S0h+l(n + 1)/0h(n) (2.4.1)

H~(C) : SUh+I(T~ -~- 1)/Uh(n) (2.4.2)

H~(K) ----- Spa+l(n + 1)/(Sph(n)• (2.4.3)

from the obvious U'(t, F) : U*-'(t, F). As with elliptic spaces, H~ (F) is a complete globally symmetric isotropic

pseudo-RIElVL~NNian manifold M ~ , which is the hyperbolic space over F in case h = 0; it is simply connected if F ~ It, and H~(R) is the manifold H~/{=t= I} of [17].

2.5 . Notation for some LIE groups. The compact simply connected LrE group of CART~ classification type X will be denoted X; thus A m = SU(n-}- 1), B.(n > 1) is the universal covering group Spin(2n + l) of S0(2n + 1), Cn = Sp(n), D~ -~ Spin(2n) for n > 1, and G~ is the group of sutomorphisms of the CAYLEY-DICKSON algebra Cay. In general, the CLIFFORD algebra construction gives a double covering group Spin h (n) of S0 a (n). Finally, F~ will denote the connected LIE group of type F4 whose maximal compact subgroup is B 4. F4* is centerless and simply connected.

2. 6. The indefinite CAYLEY planes P~ (Cay) and H~ (Cay). Recall that the isotropie RIE~AN~ian manifolds are the Euc~,IDean spaces, the spheres, the elliptic and hyperbolic spaces over a field F, and the manifolds F4/B4 and F*/B4 in invariant RIEMAN~ian metric. The latter two, the CAYT,EY elliptic and hyperbolic planes, have indefinite metric analogs; furthermore, each ob- viously gives a negative definite pseudo-RIEMANz~ian manifold on reversal of its metric.

Let a and ~ be the automorphisms of F4 defined by a = ad (s) and ~ = ad (t)

3 2 J O S E P H A . W O L F

where s ~ B, c F4 is the unique central element of order 2 in B4, and where (-I.

t c F, is the element of ]]4 with image \ 0 ~ 1/ in SO (9). There is a direct

sum vectorspace decomposition ~4 ~ ~Ba q- ~ where !B4 is the eigenspace of

q- 1 for a a n d ~ff~ is the eigenspace of - 1, and ~ ! B 4 q - ~ / - 1 ~ . As aT:----va, we also have ~B4=~1 q- ~1 and ~ g ~ = ~ - k ~ where.~, is the eigenspace of + 1 for ~ and ~ , is the eigenspace of - I; ~ - - - - - ! B ~ - ~ '

is the LIE algebra defined by ~B~ ---- -~1 q- ~ 1 ~1 and ~ ' ---- ~ + ~ / ~ 1 ~ and a induces an automorphism of ~ for which ~ is the eigenspace of q- 1 and ~{R' is the eigenspace of - 1 . Similarly, applying ~ to the decomposition

of ~*, we obtain a LIE algebra ~ ~- ~ -{- V ~ 1 ~ ' . To simplify the above, we first observe that !B~ ---- ~piul(9) from ~4 ~ ~piu(9) and the definition

tp of 3. Let G' and G" be the adjoint groups of ~ and ~4, and let H ' and H" be the corresponding analytic subgroups for ~ . G' and G" are real forms of F4, and are noncompact because H ' and H" are not compact. H ' and H" each has maximal compact subgroup of type D4; thus the maximal compact subgroups of G' and G # have dimensions > 28 ; it follows that G' ----- G" ---- F*

n and ~ - - - - ~ - - - - ~ . Now it is clear that we have only: ~ 4 ~ - - ~ 4 ~ [ ~ ,

~ * : ! B ~ - ] - ~ t * and ~ * : ~ - k ~ ' where ~ * ----- ] / / ~ l O J l , ~I~' is given above, and !~ ---- ~piu ' (9) .

The decompositions above yield symmetric coset spaces F,/B4, F*/B, and F*/Spin~(9). With KILLr~G form for invariant metric, we obtain spaces of non-positive involutive curvature with metrics respectively of signature (16,0), (0,16), (8,8); the statement about (8,8) comes from the equivalence of ~rt' and

~ / ~ 1 ~ ' , which comes from the equivalence of F~ and F~ above. The CxYL~u elliptic planes are the following symmetric coset spaces with

pseudo-R~E~zq~ian structure of non-negative involutive sectional curvature induced by the negative of the K~r,~zs~ form :

p2 (Cay) = F,/Spin (9),

P~ (Cay) ---- F~/Spin ~ (9),

p2 (Cay) -~ F~'/Spin (9),

positive definite CAYLEY elliptic plane.

indefinite CAYLEY elliptic plane.

negative definite CAYL~.u elliptic plane.

(2.6.1)

(2.6.2)

(2.6.3)

The CAYLEY hyperbolic planes are obtained the same way, with non- positive involutive sectional curvature from the KILLING form:

H~(Cay) ---- F*/Spin (9), positive definite CAYLEY hyperbolic plane. (2.6.4)

H 2 (Cay) ---- F~/Spin 1 (9) , indefinite CAYLEY hyperbolic plane. (2.6.5)

H2 ~ (Cay) ~- F,/Spin (9) , negative definite CAYLEY hyperbolic plane. (2.6.6)


P~ (Cay) and H~ (Cay) are complete because they are globally symmetric; each is a pseudo-RIE~ANNian manifold M ~ , and it is known from the theory of R~MAN~ian symmetric spaces that they are isotropic for h ye 1. We will now prove that P~(Cay) is isotropic; isotropy then follows for H~(Cay). To do this, we must show for every real number y that Spin 1 (9) is transitive on the nonzero vectors of square norm ~ in ~j~' -~ R~ 6.

Let a be a CARTAN involution o f ~ which preserves ~piul(9) ; ~ff~ ~ 9~ ~ !B under a where 9~ is positive definite and !B is negative definite. Spin(8), the subgroup of Spin~(9) which preserves this decomposition, maximal compact subgroup of Spin1(9), acts by u : a -~ b --->/(u) (a) + / ( t ( u ) ) (b) where /: Spin(8)--> SO (8) is the projection and t is triality. I f a r 0 r b, it follows that the isotropy subgroup of Spin(8) at a ~ b lies in the fixed point set Gs of t. Given ~ > 0 > f l let S ~ : (a~9~:I[a[I ~ - a } and S ~ : { b ~ : [ 1 5 [ ] 2~-fl}; it follows by counting dimensioes that Spin (8) is transitive on Sa • S~ and G2 is its isotropy subgroup 2~ . Finally, let (gt} be a 1-parameter group in Spin1(9) such that a(gt) -~ g-i 1 . A s each g, is a semisimple linear transformation of ~ff~' with all eigenvalues real, and each g~ preserves the inner product on ~l~' ----- I~ 6, we can replace the parameter by a multiple and find r E ~[ and s E!B such that I[r[I 2 : 1 ~-- -- []s[[ ~, gt(r) : cosh(t)r ~- sinh(t)s, and g~(s) = sinh(t)r ~- cosh(t)s.

Let x E ~ ' , x : / :0, Itxl[ ~ = ~ ; x ~ - - a ~ - b w i t h a E g ~ a n d b ~ ! ~ ; I ] a [ [ 2 = ~ and lib ]]2 : ft. Applying an element of Spin(8), we may assume

a : V ~ r and b : V - ~ f l s .

/cosh(t) sinh (t)~ Now a close look at R~ and matrices \sinh (t) cosh (t)] shows:

(1) I f ~, :>0, then some gt carries x to i ~ r .

(2) I f ~ 0 , then some g, carries x to r ~ -s .

(3) I f ~ ~ 0 , then some gt carries x to ~ / ~ - ~ s . This completes the proof that P~(Cay) is isotropic.

We have proved that each of the CAYLEY planes is an isotropic pseudo- l~EMA~ian manifold. Observe also that each CA~L]~u plane is complete, symmetric, and simply connected.

2.7. CAnTaN'S technique for the full group of isometrics of a R~EMX~ian symmetric space can be extended to indefinite metric. We will make that extension now, and then calculate the full group of isometrics for each of our model spaces in w 2.8.

2) I a m indebted to Professor JACQUES TITS for bringing to m y a t ten t ion this fact t ha t Spin (8) is t ransi t ive on S T • S~.

3 CMH vol. 39

34 JosEP~ A. WOLF

Le t M~ be an irreducible complete connected s imply connected pseudo- RIE~ANNian symmetr ic manifold, let s be the s y m m e t r y at x c -~/~ ; let G ----- -~ I o (M~) and G' ~ I (M]) , and let H and H ' be their respective i sot ropy subgroups at x. The problem is to find G' and H ' f rom G, H and s. G' = G- H r because M ] is connected, so i t suffices to find Hr ; knowing the subgroup H" ---- H ~ s . H of H ' , then, it suffices to find (k I = 1, ]c 2 . . . . } c H ' such tha t H ' is the disjoint union of the k~H". The whole point is t h a t the following gives enough informat ion abou t the k~ so that , in any par t icular case, one can find them:

Theorem. Each k i, i ~ 1, induces an outer automorphism o/ H which is induced by an automorphism o/ G. The automorphisms o/ H induced by k~ and kj di//er by an inner automorphism o/ H, if and only i/ k~ ~ k s.

Le t a--~ ad(s) ; t hen (~-----.~ + ~1~ where .~ is the a-eigenspace of ~-1 and ~J~ is the eigenspace of - 1. As M~ is irreducible, i.e., as ad (H ) is irreducible on ~ , the centralizer of ad (H) in the algebra of linear endomorphisms of 2R is a real division algebra F (by SCHVR'S Lemma) . Now ident i fy H , H" and H ' with the l inear groups t hey induce on ~[}~. I claim th a t we need only prove F' c H" . F o r then if k' E H ' induces an inner au tomorph i sm of H , so /c' k e F for some ]c c H , we have /c'/c ~ F r because (since it preserves the metric) det . (k' k) ~- • 1, whence k' k E H" and so /c' e H" ; the theorem would follow.

The fact t ha t F ' c H" is obvious i f F = R (in which case 1 E H i s 1 o F ' and s ~ H " is - l e F ' ) ; i f F r i t is a consequence of:

Lemma. (1) F ~-- R i /and only i/ H is semisimple. (2) F :/: K . (3) These are equivalent: (a) the center o/ H is a circle group, (b) H is not semisimple, (c) M~ has a G-invariant inde/inite metric K~HLER structure which induces the original pseudo-RIEM~Nian structure, (d) F ~-- C.

Remark . This is the indefini te-metric version of ([20], L e m m a 2 .4 .3 ) , which is essentially due to CXRTAN [5, 6] with par t of the proof t aken f rom CXaTAN [5] and BOREL [3].

Proo/o/Lemma. Let Z be the center of H ; t hen Z ~ F ' because Z c F and det. z ----- 4- 1 for eve ry z e Z . Thus F :fi (~ implies t h a t Z is finite. H is reduct ive because i t is an irreducible linear group. I t follows tha t H is semisimple if F ~: C.

Suppose t ha t F :/: R, i.e., t h a t F ' has an e lement J wi th J~ -~ - I . Then

we have real subalgebras ~ _ ~ . ~ c + ( i + 1//-- 1 J ) ~ and ~ = . ~ a ~ _

-~ (I -- ~/---- 1 J ) 2R of the complexified algebra ffi c which are complex conjugate over ffi, which span (~a over R, and which have intersect ion ~ o . As H is connected (because M ] is s imply connected), this is A. F~O~CHER'S criterion ([7], w 20) t h a t J define a G-mvar ian t complex s t ruc ture on M~ -~ G/H. F r o m our p s e u d o - R I ~ . ~ i a n metr ic we now have a G-invar iant indefinite hermi t ian


metric on M~ ; the complex s t ruc ture is parallel because H contains the holon- omy group, so the indefinite hermi t ian metr ic is KXn~LERian.

Continue to assume F ye R. Le t T be a maximal compact subgroup of H , so T" : T ~ s �9 T is maximal compact in H" ; ex tend T to a maximal compact subgroup S of G, so S " ~ - S ~ , s . S is maximal compact in G " - - - - G ~ s . G . The group G is semisimple ([15], p. 56), because M~ is not flat as a consequence of F :/: R . Thus ffi" has a CXRTAN involut ion v which induces a CARTAN involution of 9 " , and we m a y assume ~ tr ivial on S" and T" . In part icular , z commutes with a ---- ad (s) and preserves the summands in ffi ~- 9 -k ~ . The t ransformat ion t of !l~ induced by ~ normalizes the linear group H , so tF't -1 ---- F ' . I f F ~- K , then t centralizes F because F' has no outer automorphism and its e lement of order 2 is central . I f F : C and t does not centralize F , then t / t - l -~ ] for every / ~ F . Now let 9 ~ - - 9 + - k 9 - and

: ~I~+ -k ~I~_ where subscripts denote the eigenspace of -k 1 or - 1 for

3, and we define 9 * ~ - 9 + ~ - V ~ 1 9 - , 9 : R * - - ~ 0 ~ + + V " ~ 1 ~ _ a n d f f i * : ---- 9 " -k ~ * . I f G* is the adjoint group of (~* and H * is the analyt ic subgroup for 9 * , then G*/H* is a RIEMANNian symmetr ic space. I f t centralizes F, then J preserves bo th ~ + and ~[R_, and thus induces a t ransformat ion J * of square - I on ~ * which commutes with the irreducible linear group H*( - - - -ad(H*) ) on ~[R*. Otherwise, J interchanges ~ + and 9~_. Then J

induces a t rans format ion J ' of ~0~* by J ' (X) ---- ] , /~ 1J (X) if X E ~ + or if

X ~ V ' ~ 1 ~ . J ' centralizes H * and has square - k I . This is impossible because H * is irreducible on 93~*. Thus J preserves bo th ~ + and ~[R_, and induces J * . I t follows, as with J on G/H, t h a t J * induces a G*-invar iant K~ItLER metr ic on the compact R I ~ M A ~ i a n symmetr ic space G*/H*. Now the cohomology group It*(G*/H*; R) :/: 0 so the h o m o t o p y group ~2(G*/H*) is infinite, and the h o m o t o p y sequence

0 : ~(G*) --> z~(G*IH*) -> ~I(H*) -> ~I(G*) : finite

shows tha t :~ (H*) is infinite. Thus H*, and consequent ly H , is not semisimple. We have proven t ha t H is no t semisimple when F :/: R , and t h a t M~ has

a G-invariant indefinite metr ic KX~aLER s t ruc ture then which induces its original pseudo-RIEMAN~ian s t ructure . As H is reduct ive, it also follows t h a t the center Z of H is infinite; as Z c F ' , i t follows t h a t F ---- 0. The L e m m a now follows, and the Theorem is proved. Q . E . D .

2 .8 . Enumera t ion of the full groups of isometrics of the model spaces. I t is known (V17], w 3) and easily seen t h a t the full group of isometries I(R~)

is the semidirect p roduc t 0 t'(n) �9 R ~ acting by

(A, ~) : ~--> A(~) q-

36 JOSEPH k. WOLF

where A ~ 0 h (n), a c R ~ = underlying vector group of R~, and x ~ R~; the action of A is the usual linear action.

I f h < n - 1, then the universal pseudo-RIEMA~ian covering manifold S~ of P~(R) is the quadric S~ : {~ e R~+I : [[~I[ 2 = 1} with induced metric; if h----- n, then ~ is one of the two components of S~; if h ~ n - 1, then the situation is rather complicated ([17], w 11) and ~ is an infinite covering manifold of S~. Similarly, the universal pseudo-RIEMAN~ian covering manifold ]t~ of H~(R) is described by the quadric H~ : (~ ~ ~th+l :vn+l 1 [ . ~ [ [ 2 : _ _ 1} with induced metric. I t is known ([17], w 4.5) that I(S~) ----- 0a(n + 1) and I(H~) ---- ---- 0a+l(n + 1). Together with ([17], w 11), this will suffice for our purposes.

We will apply the technique of w 2.7 to the other model spaces. The starting point is a result of K . N o m z v ([15], Th. 16.1) which assures us that the representations M ------ G/H of model spaces M of nonconstant curvature, given by ( 2 . 3 . 2 - 3 ) , (2 .4 .2--3) and ( 2 . 6 . 1 - 6 ) , have the property: Io(M ) consists of the isometrics induced by G. In each case, this means I 0 (M) ---- G/Z where Z is the center of G ; Z is trivial if G is of type F4, Z = { 4- I} if G = Sp a (n + 1), and

Z consists of the scalar matrices exp (2a l / - - lm/(n + 1))I if G = SUk(n + 1). Let M be a CAYLEY plane G/H as in (2 .6 .1--6) . Then G - - Io (M ), q con-

tains the symmetry, and H = Spina(9) has no outer automorphism. By Theorem 2.7, G = I (M) :

I(p02(Cay)) = I(H~(Cay)) = F4 ; Spin (9) is isotropy subgroup. (2.8.1)

I(P~(Cay)) = I(H~(Cay)) = F~; Spin ~ (9) is isotropy subgroup. (2.8.2)

I(P2~(Cay)) = I(H~(Cay)) ---- F*; Spin (9) is isotropy subgroup. (2.8.3)

Let M be an indefinite quaternionic elliptic space G/H; G = Spa(n + 1) and I0(M ) = G/(4-I}. I f n > 1, then Spa(n) is not isomorphic to Sp(1), and it follows that H/(4- I} has no outer automorphism. I f n = 1, then G is of the type C2 ; CARTA~ type C~. is the same as type B2 so one would expect that M have constant curvature; indeed, it is standard that P01(K) is isometric to the sphere S 4 and easily seen that P~ (K) is real hyperbolic space H ~ with metric reversed. We set aside these spaces of constant curvature. Now G contains the symmetry, so Theorem 2.7 yields:

I(P~(K)) = I(H~,_h(K)) = Spa(n + 1)/(4-I} when n > 1; (2.8.4)

(Sp a (n) • Sp (1)}/{ 4- I} is isotropy subgroup.

Let M be an indefinite complex elliptic space G/H; G = SUa(n + 1) and Io(M ) is SUa(n + 1)/{scalars}. The image of H in I0(M ) is isomorphic to U a (n); its only outer automorphism, the usual conjugate-transpose, is induced


by the automorphism ad(~) of G where c~ is the isometry of M induced by conjugation of 13 over R. As G contains the symmetry , Theorem 2.7 yields:

I(P~(C)) ~-- Io(P~(C)) ~ ~ . Io(P~(C)) : I(H~_h(C)), (2.8.5)

where c~ is induced by conjugation of C over R; Uh(n) v cr Uh(n) is isotropy subgroup.

We have now determined the full groups of isometrics of the model spaces.

2 .9 . Sectional curvatures of the model spaces are not well behaved, except in the cases of definite metric and constant curvature:

Theorem. Let ( M , Q ) be a p seudo -R iRM~r ian mani/old P~(F) or H~(F) with F = t3 or K and 0 < h < n , or P~(13ay) or II~(13ay). Then the sectional curvatures o/ (M , Q) are not bounded and do not keep one sign.

Proo/. L e t G ~-- I o (M, Q) and let H be the isotropy subgroup of G at x E M . Then G is semisimple, .~ is the eigenspace of q- 1 for the involutive isometry of (~ induced by the symmet ry to (M, Q) at x, ffi ~ .~ ~ 9~ where 9J~ is the eigenspace of - 1 for T and both summands are stable under a Ca_RTA~ involution a of ffi, and a nonzero real multiple of the Kn~LING form B of ffi induces Q through its restriction to ~02. Thus we are in the si tuat ion of Theorem 1.3.1 . I f one changes Q to a . Q, he changes a sectional curvature K ( S ) to a-XK(S) ; thus the assertion on P~(F) or P~(Cay) will follow from the corresponding assertion for II ;_h(F ) or II~ (13ay), and we m a y assume tha t Q is induced by B. As [~[R, ~02] c ~ , we have seen in the proof of Theorem 1.3.1 tha t a nonsingular 2-plane S c !ff~ with basis {X, Y} has sectional curvature

K ( S ) = IIXII2 II[X'Y]II2 �9 [ I y I I ~ - - B ( X , y ) ~

where I l Z l l ~ denotes B ( Z , Z ) . We also write (fi = (fi+ q- ~ _ , ~ = ~}2+ q- ~J~_ and ~ = .~+ q- ~_ where subscripts denote the eigenspace of -[- 1 or - 1 for the C~TAN involution a, and we write Z = Z+ q- Z_ for every Z c (fi where Z+ ~ ~+ = .~+ -[- RI2+ and Z ~ I~i_ = ~_ q- ~J~_. B is positive definite on ~ _ and negative definite on ~i+.

Choose X E ~1~_ with I[ X ll ~ = 1 and let U be the subgroup of H consisting of all elements which preserve ~ + and ~R_. U is a maximal compact subgroup of H and 1I ----- ~+. Le t V be the isotropy subgroup of U at X. Transi t iv i ty of H on the light cone I] WI[ ~ = 0 r W in ~ implies t rans i t iv i ty of V on the unit sphere S+---- { W , ~ I I + : I I W I I 2 = - 1 } in 9Y~+. I f ( M , Q ) - ~ H ~ ( F ) , then ~IR carries the s tructure of F~ and H acts as Ua(n, F) �9 F ' . We then define a subspace 0 :/: ~R = X • ~ X . F c ~IR_ which is nonzero because F ~: R. Let S_ be the uni t sphere {W~9~: I]WII 2 = 1) in ~R; i f F = 13 then S_ is

38 JOSEPH A. WOLF

two points; if F = K then V is t rans i t ive on S_. I f (M, Q) - - H~l(Cay), we define 9~ ~-- X • ~, ~I~_. Then U ~- Spin(8) ; let t be its t r ia l i ty au tomorphism and let / : Spin(8)-+ SO (8) be the project ion; if SO (8) acts on 2R+ and !IR_ b y its usual act ion on R a, t hen U acts on ~ = ~(R+ (~ ~ _ by u --> f (u) G f ( t ( u ) ) ; then the action of V on ~t is the usual act ion of S0(7) on R 7 , so V is t ransi t ive on the uni t sphere S_---- (Wc~R:IIWII 2 - - 1} in ~t .

F r o m t rans i t iv i ty of V on S+, and on S_ when the la t ter is connected, we see tha t there are real numbers a and fl such t ha t K (S ) ---- a and K ( T ) ~ ff S is spanned b y X and some Y ~ S_, and if T is spanned by X and some Z ~ S+. Now ~ :~ fl because (M, Q) is no t of cons tant sectional curvature .

Choose YE S_ and Z E S+; for eve ry real nu m b er b, let b' ----- l / N + 1, define W b = b Y + b ' Z , and let S b be the plane spanned by X and W b.

We have X_L Wb and I]Wb[12~ -- 1. Thus K ( S b ) = -- I t[X, Wb][[~= -- {]I[X, bY]I[ 2 --[- I][X, b'Z][I ~} ~ - {b2~ - (b ~ + 1)fl} = b~(fl - ~) ~- fl,

which is unbounded for [b [ large. To see t ha t sectional curva tu re does not keep one sign, we can ei ther app ly Pa r t 3 of Theorem 1 .3 .1 or observe t h a t the plane T b spanned by X and b ~ Y -~ b Z has K ( T b ) -~ ~ Jr- b~(~ - fl). Ei ther way, Theorem 2.9 is proved. Q . E . D .

Remark . One can show ~ ----- 4fl < 0 above.

3. Groups transitive on cones and quadries

The first step in our s tudy of isotropic spaces is:

3 .1 . Theorem. Let G be a closed connected subgroup o/ S0 h(n) which is

transitive both on a component L of the light cone {x E R~ : [I x ll ~ -~ O, x :/: 0} and a component Q of a non -emp ty quadric {x ~ R~ : [[ xt[ 2 = a} (a ~ 0). T h e n

(a) (7 = S 0 h ( n ) ; o r

o r

o r

o r

(b) G ---- SUh/2(n/2) or Uh/2(n/2) ;

(c) G = Sph/4(n/4) or Sph l ~ (n / 4 ) .T or Sph/4(n/4) �9 Sp(1);

(d) G : Spin(9) with n = 16 and h - ~ 0 or 16; or G = Spin~(9) with

n - ~ 16---- 2 h ;

(e) G - - S p i n ( 7 ) with n - ~ 8 and h = O or 8; or G----Spin3(7) with

n : 8 - - - - - 2 h ;

Isotropic manifolds of indefinite metric ~9

o r

(f) G = G~ wi th n = 7 and h = 0 or 7 ; or G = G* wi th n ----- 7 and h = 3

or 4 .

Here G* denotes the noncompact centerless real group o] the type G 2 .

Conversely, every group listed above is t ransi t ive on the nonzero elements of a n y

given square norm i n R~.

R e m a r k . Our proof is based on the case h = 0, which is the celebrated MONTGOMERY-SAMELSON-BOREL classification of groups t ransi t ive on spheres ([12], [1], [2]; see w 27 of [16]).

3 .2 . Proo/ . I f n = 2, then dim. Q = 1 = dim. S0h(n) and necessarily G : S0h(n). NOW assume n > 2; then G is irreducible on R~ ([17], L e m m a 8.2). Now G is reduct ive and its connected center e i ther is tr ivial or is a circle group which gives a complex s t ruc ture to R~; it follows ([14], Th. 6) t h a t fli is invar iant under a CARTAN involut ion a of ~ h (n) which induces a CXRT~ involut ion o f f l i , a----1 if h = 0 or h = n .

Le t x ~ Q. The isot ropy subgroup H of S0h(n) a t x is reduct ive, algebraic and connected, and thus ([14], Th. 6) .~ is invar iant under a C~TAN involut ion v; ~ and ~ are conjugate in the group of inner au tomorphisms of SOb(n), so wc m a y change x and assume a ---- 3. Now ~ (n) = ~ + ~ where ~ is the eigenspaee of ~- 1 for a and ~ is the eigenspace of - 1 ; fli ---- fliK ~- flip and .~ = ~K ~- ~P where subscripts denote intersect ion with ~r or ~ . Define

~ h ( n ) * - - - - ~ + V~-- 1 ~ , fli* : fli~ + V---- 1 flip and .~* : . ~ + V~-- 1 .~p; then ~ h ( n ) * - - - - ~ ( n ) and ~ * : ~ ( n - 1). Le t G* be the analyt ic subgroup of S0(n) corresponding to fli* c ~ ( n ) . Trans i t iv i ty of G on Q implies fli ~- .~ = ~ h ( n ) ; thus fli* ~- .~* : (fli ~- .~)* = ~ ( n ) and it follows t h a t G* is t ransi t ive on the sphere S ~-1 = SO(n)/SO ( n - 1 ) .

According to MO~TGOMV.Rr, SAMELSO~ and BOREL, t rans i t iv i ty of G* on S n-1 implies

(a) (7* = S O ( n ) ; o r

o r

o r

o r

o r

(b) G* : SU(n/2) or U(n/2) ;

(e) G * = Sp(n/4) or S p ( n / 4 ) . T or S p ( n / 4 ) . S p ( 1 ) ;

(d) G* = Spin(9) wi th n ----- 16 ;

(e) G* : Spin(7) with n ---- 8 ;

(f) G* = (~2 wi th n = 7 .

4 0 JOSEPH A. WOLF

Because G c S0h(n), cases (a), (b) and (c) above yie ld the corresponding possibilities for G in the Theorem; t hey h a v e the required t rans i t iv i ty . Now we need only examine (d), (e) and (f) above when 0 < h < n . We will need the fac t [13] tha t , b y t r a n s i t i v i t y of G on L , a m a x i m a l compac t subgroup M c G is t r ans i t ive on a com pac t de fo rmat ion r e t r ac t of L , and t h a t this r e t r ac t is a componen t of S h-1 • S T M because the full l ight cone is homeo- morph ic to S h-1 • R • S ~-h-1. As M preserves c o m p l e m e n t a r y posi t ive definite and nega t ive definite subspaces of R~ ([18], p. 79), we m a y t ake S T M and S a-~ to be the respect ive uni t spheres in these subspaces. We m a y assume h _< n -- h.

Le t G* --~ Spin(9) wi th G n o n c o m p a c t and 0 < h ~ n - h. Then n ~-- 16, G - ~ Spink(9) wi th 1 ~ k ~ 4 , and M i s a quot ient of Spin(k) • S p i n ( 9 - k ) . N o w k ~ 1 because a s imple fac tor of M is t rans i t ive on S T M and n -- h - - 1 ~> 7 ; thus h ~-- 8 and we are in case (d) of the Theorem. The required t r ans i t i v i t y of G was seen in our p roof t h a t p2 (Cay) be isotropic.

Le t G*- -~Sp in (7 ) w i t h G n o n c o m p a c t a n d 0 < h _ ~ n - h . Then n~-- 8, G ~ Spink(7) wi th 1 ~ k ~ 3, a n d M i s a quot ient of Spin(k) • S p i n ( 7 - k ) . I f k ~- 1, then Spin(6) is t r ans i t ive on Sn-h-~; n -- h -- 1 ~_ 3 then implies n -- h -- 1 ---- 5 and h - 1 ---- 1, so the i so t ropy subgroup Spin(5) of Spin(6) on S 5 is t rans i t ive on S 1 ; t h a t is impossible. I f k ---- 2, then Spin (5) is t r ans i t ive on S T M because Spin(2) is no t ; thus n -- h -- 1 ~ 4, so h -- 1 ---- 2 and M ' x Spin (2) is t rans i t ive on S ~ where M ' is i so t ropy of Spin (5) on S4; t hen M ' ~-- Spin (4) implies t h a t M has a 2-dimensional subgroup which acts t r ivial ly on all of R~; t h a t is impossible. N o w k ---- 3, so M is locally (Spin(3))3 ; thus h ----- 4, two fac tors of M act ing as SO (4) on one S s and the th i rd act ing s imply t rans i t ive ly on the o ther S 3, as in case (e) of the Theorem.

L e t G*----G2 w i t h G n o n c o m p a c t a n d 0 < h _ ~ n - h . Then n - ~ 7 and M is a quot ient of SU (2) • SU (2). I t follows t h a t h ~ 3, M ~ SO (4), G is the n o n e o m p a c t centerless group G2* of t ype G~, and we are in case (f) of the Theorem.

3. 3. I n order to complete the p roof of the Theorem, we m u s t find act ions of Spin s (7) on R~ and G~ on R~ wi th the desired t r ans i t i v i t y propert ies .

We have Spin(7) c S0(S) t rans i t ive on the uni t sphere S 7 c R s and wi th i so t ropy subgroup G2 a t x ~ S 7. The t h e o r y of R I E M A ~ i a n s y m m e t r i c spaces provides an e l emen t t E G2 of order 2 whose central izer K 1 in G2 is iso- morph ic to S0 (4). Le t A and B be the eigenspaces of Jr 1 and of - 1 for t on R 8 .

W e first p rove dim. A ~ 4 ---- d im. B . B has even dimension 2b, l ~ b ~ 3 , and we m u s t p rove b ----- 2. L e t y be an e lement of the uni t sphere S 6 in the


tangentspace (SV)x. G2 is transit ive on S 6 with isotropy subgroup J ~ SU(3) at y . As rank. J ----- 2, we m a y assume y chosen so t ha t t �9 J . We ident ify (S7)x with x L by a G~-equivariant parallel translation; now x and y span a plane T in A, and J acts on T 1-. Thus i f b =/= 2, t would be - I on T • and thus would be central in J , or would represent an element of U(3) not in SU(3) and thus not be in J . This shows b -~ 2.

We will produce groups acting on R ~ = A q- ~ I B and R ] - - A ' q -

-k ~/--~l B, where A ' = A ~ x • Le t v be the automorphism ad(t) of ~pirt(7); then ~pirt(7) --~ R + ~ where ~ is the eigenspace of + 1 for and ~ is the eigenspaee of - 1 , and ffi 2 : R ' - k ~ ' where ~'----ffi 2 ~ ,

and R ' - ~ ( b 2 ~ ~{ is the algebra of K 1. Now ~2 : R' + V - 1 ~ ' is the

LIE algebra of G*. ~{ + ] / ~ 1 ~ is ~pirt3(7) because it is contained in ~D4(8) where 804(8) acts on R~, and because S04(8) cannot contain Spin(5). Thus G* acts on R 7 and Spin3(7) acts on R~.

Given ~ > 0 = > f l , we define 3-spheres S~-~ { w e A : ] l w ] l 2--or and

S#----- ( w � 9 B: [[w]l a-~fl} and a 2-sphere S ' ~ - S ~ A ' . K 1 acts effectively on A' (~ B and is isomorphic to 80(4); thus we view K~ as the image of a representation [ Q g of S0 (4) on A' Q B. g is faithful because [ Q g is faithful. I f / is no t trivial, then [ : SO (4) --> S0 (3) is onto and it follows from the local product s t ructure of S0 (4) tha t K~ is transit ive on S~ • S~. We will prove tha t [ is not trivial. Choose nonzero z E B, let G 2 be the isotropy subgroup of Spin(7) at z, let e be the central element of order 2 in Spin(7), and define t' ~-- re. Now e acts as - I on R s by irreducibility of Spin(7), so t' = - t on R s and t' �9 G 2. Any two elements of order 2 in G, being conjugate, Spin(7) has an element h such tha t hG, h - ~ - - G 2 and hth -1 ----- t'. Now K, ~ h K l h -1 is the centralizer of t' in G 2. h interchanges A and B because t' -~ - t; it follows tha t the group K generated by K~ and K, preserves both A and B. I f / is trivial, then K is the product SO (4) • • S0(4) of the rotat ion groups of A and B ; then rank. K ~ 4 > 3

rank. Spin (7), which is impossible. This proves / nontrivial.

We have proved tha t K1 is transit ive on S: • S~. The group K above is centralized by t and thus is contained in Spin3(7). I t is locally a product of groups SO (3), the number of factor being at least 3 because it contains K~ properly and the number of factors being a t most 3 because rank. Spin (7) ~ 3. Thus K is the group with LIE algebra R and is maximal compact in Spin 8 (7). A close look at 1, g and h above shows tha t K is t ransi t ive on S~ • S~. Now, as in the proof t ha t p2 (Cay) is isotropie, it follows tha t fl* (resp. Spin 3 (7)) is transitive on the nonzero elements of any given square norm in R~ (resp. in R~). Q.E.D.

42 JOSEPH A. WOLF

3.4. Some eoset representations for quadrics. In R~ we define quadrics

S : ( x ~ R ~ : [ I x t [ 2-~ l} and H ~ - ( x ~ R ~ : [ I x [ l 2 ~ - 1}.

Besides the usual representa t ions of S and H as coset spaces of S0h(n), U h/2 (n/2) and SU h/2(n/2), and Sp a/4 (n/4) �9 Sp (1), Sp a/4 (n/4) �9 T and Sp a/4 (n/4), we have :

1. S p in ( a ) /S p in (7 )= S c R016

2. Spin~(9)/Spin(7)-~ S c R~ 6

3. Spin(7)/G~ ~ - S c Ro s

4. Spin3(7)/G * = S c R~

5. G~/SU(3)~--S c R~

6. f l* / su~(3)= s c R~

7. G*/SL(3; I t ) = S c R3 7

Replacing the inner p roduc t by its negat ive on R~, these representat ions yield representat ions for H in R~66, R~ 6 , R88 , R~, R 7 , R 7, and R 7 .

4. Structure of locally isotropie manifolds

Theorem 3.1 enumera t ed the candidates for group of local isometries in a locally isotropic manifold M~. This is used in the s t ruc ture theorem, Theorem 4 below. I t turns out t ha t no t all the groups listed in Theorem 3.1 occur here, however; this happens in such a way t ha t we are able to extend, in Theorem 4 (4.), t he theorem tha t an isotropic LO~.NTZ manifold has constant curvature .

Theorem. Let M'~ be a connected locally isotropic pseudo-Rz~M~NNian mani- /old. Given x E M~, let G~ be the group o/local isometries at x, let ~ denote the Lz~ algebra o/germs o/ KILLIlVG vector/ields at x, and view ( ~ as the subalgebra o/ ~x consisting o/the elements which vanish at x. Then:

(1) M'~ is locally symmetric.

(2) M'~ is locally homogeneous: given x, y E M~, there is an isometry / o /a neighborhood o/ x onto a neighborhood o~ y with /(x) -~ y. In particular, ] induces isomorphisms ~ ~__ ~y, ~ ~__ (~y and G| _~ G~.

(3) M~ is characterized locally by n, h, G~, ~ , and (only necessary i/ ~ = ~* and n ----- 2h = 16) the sign o/involutive sectional curvature, in the sense that the only possibilities are:


G~ ~ M~ locally isometric to

not semisimple R] S0h(n),

case of constant curvature

U i'' (n'), 2h ~ = h, 2n ~ = n

Sph" (n") �9 Sp(1), 4h n : h, 4n ~ : n

Spin(9), n = 16, h = O or h = 16

$~h (n + 1)

~f:)h+l(n -Jr- 1)

$1I h' (n' ~- 1 )

~ h ' +l(n' -Jc 1)

~ph" (n" + 1)

~ph"+~ (n" § 1)

P~(R)

H~(R)

e~; (c)

n~;(c) nrs Ph.(K) rift Hh,,(K)

P~(Cay) or H~(Cay)

H~(Cay) or P~(Cay)

Spin~(9), n = 2h : 16 ~* P~(Cay) or H~(Cay)

(4) I / h is odd, or i/ n -- h is odd, then M'~ is o/ constant sectional curvature.

Proo/. The case n ---- 2 is trivial because there each G| = S0h(n) and M~ has constant curvature. Now let n > 2. As seen in the proof of Theorem 3.1, G. acts irreducibly on the tangentspace T . and every representation of G| is fully reducible.

Let ~ : ~ . - + T . be evaluation at x. ~ is a linear map with kernel ffi.. We have a natural representation g-> ad(g) of G. on ~x by automorphisms, and u(ad(g)X) = g ( n ( X ) ) for every g ~ G . and every X ~ . . Given z E M~, ffi z extends to a LIE algebra of KILLI~O vectorfields on a normal coordinate neighborhood U of z. We may choose z with x c U. By irreducibility of G z on Tz, now, ~x has an element which does not vanish at x. By irreducibility of G. and equivariance of u, it follows that ~ ( ~ ) = T . and in particular ~ . has dimension n + dim. G s.

G. is represented faithfully on T~ because M~ is connected. Identifying T . with R~ and G. with its linear action, the identity component G of G. is listed in Theorem 3.1. By irreducibility of ffi. on T| and by the LEvI-WKITEttEAD- MALC~.V' Theorem, ffi. is a maximal subalgebra of ~ . ; if ffi. is semisimple it follows either tha t ~ . is semisimple or tha t ~ = ~ . + gad (~ . ) ; in the latter case Rad(~.) is commutative by irreducibility of (~., whence M~ is flat in a neighborhood of x, and G. = 0 h (n).

I f G is listed under (e) (resp. (f)) in Theorem 3.1, so G is Spin(7) on I~ or Rs s or Spina(7) on R~ (resp. G~ on R~or R~or G~on R ] o r R~), then we

44 JOSEPH A. WOLF

choose X ~ T| ~ R~ with fIX If ~ ~ 0 and observe t h a t the isot ropy subgroup of G a t X is (~ or G* (resp. SU(3) or SUI(3) or the self-contragredient group of t ype SL(3, R) c S03(6)). F r o m the combined t r ans i t iv i ty properties of G and this subgroup (see L e m m a 8.3 of [17] for SL(3, R)) , it follows t ha t G is t rans i t ive on positive definite or on negat ive definite 2-planes in T~, so all such planes give the same sectional curva ture ; these planes form an open subset in the GRASSMA~TN manifold of 2-planes in Tx, and thus their sectional curva tures determine the curva ture tensor of M~ at x; i t follows t ha t M~ has cons tant curva ture in a ne ighborhood of x and so G x = O h (n). This is a contradict ion. We conclude t h a t G is listed under (a), (b), (c) or (d) in Theorem 3.1.

I f G is listed under (a), then - I ~ G~ = Oh(n). I f G is l isted under (b), (e) or (d), then - I E G c G~. In a n y case, M~ is locally symmetr ic a t x . This proves the first s t a t emen t of the Theorem.

Define two points of M~ to be equivalent if a neighborhood of one is isometr ic to a ne ighborhood of the other as in the second s t a t emen t of the Theorem. The equivalence classes are open subsets of M~ because ~ ( ~ ) ---- T~ for eve ry x. As the equivalence classes are disjoint, i t follows t h a t each is a union of components of M~. The second s ta tement of the Theorem follows f rom connectedness of M E .

We have seen t ha t G~ has an e lement t which is -- I on T| ; let 7 = ad (t). Then under ~, ~x = fl~ ~- ~ where ~ is the eigenspace for - 1 in ~ ; fix is the eigenspace for ~- 1. We have also seen t h a t M~ is fiat and G~ ----- ~--0h(n) if ~ is not semisimple; now assume ~ semisimple. Then ([14], Theorem 6) there is a CART)~N involut ion a of ~z which induces a CXRTAN involut ion of fl~. t lies in a compact subgroup of G~ because t ~ = 1; replacing t by a conjugate (this does nothing because t is central), we have a~ = Ta. ~ = ~K ~ ~p and fl~ = flK ~- flP where K denotes a-eigenspace

of -4-1 and P denotes a-eigenspace of - - 1 . Le t ~ * = ~ K ~- ~ 1 ~p and

f l * ~-- flK ~- V - 1 flip; then ~ induces an involut ive au tomorph ism of ~* wi th f l* as fixed point set. Le t L* be a (necessarily compact) connected group wi th LIE algebra ~ * , and let G* be an analyt ic subgroup corresponding to f l * .

Recall ~ = fl~ -~ ~ under 7; ~ : ~ - ~ T~ is an equivar iant isomor- phism, so ad(Gx) has an ( n - 1)-dimensional orbi t on ~Ll~. Le t ~ * =

= (~}~ ~" s -4- V " ~ 1 ( ~ z ~" ~ P ) ; t hen s = f l* -4- YJ~* under 7, and i t follows t h a t ad(G*) has an ( n - 1)-dimensional orbi t on !ill*. In o ther words, the compac t R ~ M ~ i a n symmetr ic space L*/G* has rank 1. B y Theorem 3 and our above el imination of (e) and (f), i t follows t h a t the iden t i ty componen t of G~ is (a) S0~(n), (b) Ua/~(n/2), (e) Spa/~(n/4) �9 Sp(1), or (d)


Spin (9) or Spin 1 (9), and that ~ is given respectively as an algebra of type (a) B~/~ i f n is even, D(~+~/~ i fn is odd, (b) A,/~, (c) Cc~14~+~, or (d) F4.

I f G~ = S0n(n), then M~ is of constant curvature, and thus is locally isometric to E~,P~(R) or H~(R). If G ~ = Ua'(n ') where 2h'-----h and 2n' = n, then ~ is either ~ l I a'(n ~ -~ 1) or ~lIh'+l(n ~ ~- 1) because it contains Ii h' (n') and is of type A~, ; thus M~ is locally isometric to P~:(C)

n ~ or to Hh,(C). I f G~ = Sph"(n") �9 Sp(1) where 4 h " = h and 4n"----n, then ~ is either ~pJ"(n" ~ 1) or ~pa"+l(n" ~- 1) because it is of type C,~,+~ and ~ph" (n" )O~ p(1 ) is a maximal subalgebra; thus M~ is locally isometric

n t t t i l t either to Pa,,(K) or to Hh,,(K ). I f G~ is Spin(9) or Spin1(9), then ~ is ~4 (only when G~ = Spin(9)) or ~ , and M~ is locally isometric to a CAYL~Y plane.

We have just completed the proof of the third statement of the Theorem. The fourth statement follows at a glance. Q . E . D .

Theorem 4 allows us to formulate:

Detinition. Let M~, be a locally isotropic p s e u d o - R i f l e m a n manifold and let S be a nonsingular tangent 2-plane at x e M~, so S is sent to a nonsingular tangent 2-plane ~, (S) at ](x) ~ N'~ under an isometry ] of a neighborhood of x onto a neighborhood in

N~ = ItS, P~'(lt, C, K or Cay) or H~'(It, C, K or Cay) .

Then S is involutive, and the sectional curvature K(S) is an involutive sectional curvature, if and only if either N~ = E~ or / , (S) is involutive on N] .

5. The universal covering theorem

Our global classifications will depend on the following consequence of Theorem 4:

Theorem. Let M'~ be a connected pseudo-RzEM~N~ian mani/old. Then M~ is complete and locally isotropic, if and only i / i t s universal pseudo-RxBM~ian covering manifold is (a) R'~, S~ or tI~; or, i/ h'-----h/2 and n ' = hi2 are integers, (b)Pa,(C) or tt~:(C); or, i/ h " = h/4 and n " : n/4 are integers,

net ~n (c) Pa,(K) or Ha,(K); or, i/ n -- 16 and h It1 = h/8 is an integer, (d) Par,2 (Cay) or H~#/(Cay).

Remark. ~'~ and I ~ are the universal p s e u d o - R I E ~ N i a n covering manifolds of P~(R) and H~(R).

Proo]. I t suffices to prove the Theorem when M~ is simply connected, and completeness and local isotropy are immediate from the existence of a pseudo-

46 JOSEPH A. WOLF

RI~MAN~ian covering by one of the manifolds listed. Now assume M~ complete, s imply connected and locally isotropic; we must show t h a t it is isometric to one of the manifolds listed.

Choose x �9 M ] . By Theorem 4, there is an isometry f of a neighborhood U of x onto an open subset V of a manifold N] in our list. The curvature tensors are parallel on both M~ and N], for those manifolds are locally symmetric . The torsion forms vanish because we are dealing with LEvI-CIVITA connections. The proof of Theorem 5 is now identical to the proof of the case of constant curvature ([17], Theorem 5): one uses N.HICKS' extension ([9], Theorem 1) of A~BROSE'S Theorem to extend / to a global isometry by extending it along broken geodesics. Q . E . D .

6. A partial classification of complete local ly isotropic mani fo lds

As in the case of constant nonzero curvature [18], one can classify the non- flat complete locally isotropic manifolds in certain signatures of metric:

6 .1 . Theorem. Let My be a complete connected locally isotropic pseudo- R I E M ~ i a n manifold which is not flat.

1. I / My is o/non-negative involutive sectional curvature with 2s ~ n and s :/= n - 1, or if M'~ is of non-positive involutive sectional curvature with 2s ~ n and s =fi 1, then the fundamental group 7~1(M'~) is finite.

2. I f ~I(M~) is finite, then (a) My is of constant sectional curvature. I n [18], the global classification of

such spaces is reduced to the classical CLXPFOaD-Kx, EX~ spherical space form problem in dimension n -- s for positive curvature, and thus in dimension s /or negative curvature; o r

(b) M~ is simply connected and thus isometric to one o/ the model spaces; o r

o~/2(C)/(1 a J} where r = s/4 and t = ((n/2) + 1)/2 (c) M a is isometric to ~ s/2 are integers, ~ is the isometry induced by conjugation of C over It , and J is the isometry induced by (oi)

- - I , 0 �9 SU(2r) • Sg(2 t -- 2r) c Sg"/2((n/2) + 1);

0 /0,_ ~ - - I t - - r

n H n / 2 or (d) M s is isometric to ,,-~s/2(C)/(1, ~ J } where r -~ ((s/2) ~ 1)/2 and t ~-- ((n/2) -{- 1)/2 are integers, and ~ and J are as above.


Remark. I t suffices to prove Theorem 6.1 in the case where M~ is of non- negative involutive sectional curvature; the case of nompositive involutive sectional curvature will then follow by reversing (replacing with its negative) the metric.

6.2. Proof of Part 1. M~* is non-negative involutive curvature with 2s _~ n and s ~: n - 1. We may suppose that M~ is not of constant curvature, for the result is known in the case of constant positive curvature ([18], Theorem 1).

Now suppose tha t M~ is not covered by a CAYLEY plane; then ~ : l~a (F) -~ M~ is the universal pseudo-RIEMA~ian covering where F = C or K has degree / over R, [ m : n and /h : s. Let -/"be the group of deck transformations of n; we must prove P finite; as I(P~(F)) has only finitely many components, it suffices to prove / ' ~ Io(P~(F)) finite; thus we may assume/1 ~ Io(P~(F)). The principle fibring v2: S~+~-l-~ I~a(F ) given by

Uh(m ~- 1, F)/Uh(m, F) --+ Uh(m ~- 1, F)/Uh(m, F) • U(1, F)

induces a homomorphism r of SUh(m ~- 1) or Sph(m ~- 1) onto Io(P~(F)) ; now it suffices to prove (P- I (F) finite. ~b-I(F) is properly discontinuous because ~ is ~0-equivariant, F is properly discontinuous, and the kernel Ker. ~b is finite. ~b-t(F) acts freely because both F and Ker. ~ act freely. 2s ~ n implies 2s ~ n - ~ / - 1, and / > 1 then gives s r 1 ) - -1 . Now our result ([18], Theorem 1) for constant curvature shows that ~b-1 (/~) is finite, and so ~1 (M~) is finite.

Now suppose that M~ is covered by a CAYLEY plane P~ (Cay) ; h = 0 or 1 by hypothesis. I f h = 0, then the CAYLEY plane is compact and our assertion is trivial; now assume h = 1 and let F be the group of deck transformations of the universal pseudo-RIEMA~ian covering u:P~ (Cay) --> M~. P~ (Cay) is a coset space G/H where G = F~ is the full group of isometries and H is the isotropy subgroup at some point x. Let a be a C~TAN involution of ffi which preserves .~ and induces a CARTA~ involution of ~ ; ~ = ~ ~ ~ where is the eigenspace of ~- I for a, where ~ is the eigenspace for -- 1, and where

~ (~ ~ R ) + (.~ ~ ~) . Let 9~ be a 1-dimensional subspace of ~ , and define A =exp(9~) , K = e x p ( ~ ) , and P = e x p ( ~ ) . As G / K i s an irreducible RIEMA~Nian symmetric space of rank 1 and of noncompact type we have G = K . P and P = a d ( K ) A ; thus G = K A K . Now d i m . . ~ = 36 and d i m . ( ~ ) = 28 because.~ is of type B4 a n d . ~ is of type D4; thus ~ ~ ~ ~ 0 and we may choose 2 c .~ ~ ~ ; it follows tha t G = K H K . Define X : K ( x ) c P2(Cay) and let g~G; then g:]c~hk~ with k i ~ K and h E H , ]c~'l(x)~X, k l ( x ) r and g ( k ~ l ( x ) ) : k l ( X ) ; it follows that

48 JosEP~ A. WOLF

g (X) meets X . Now X is a compact subset of P~ (@ay) such t h a t ~ (X) meets X whenever 7 E F . As in [18], proper d iscont inui ty of F now implies t h a t F in finite. Thus ~1 (M~*) is finite.

We have now proved the first pa r t of Theorem 6.1.

6 .3 . Proof of Par t 2. Le t F be a finite group of isometries acting freely on a model space N of non-cons tan t and nonnegat ive involut ive sectional curva ture . F lies in a maximal compact subgroup K of I (N), and there is a point y c N such t h a t the s y m m e t r y to N at y normalizes K in I (N). Le t H be the i so t ropy subgroup of I (N) a t y ; then L ---- H ~ K is a maximal compact subgroup of H , and / ' acts freely on K / L . The possibilities for N are P~ (Cay), l~a(K ) and l~a(C ) ; t h ey all have the p rope r ty t h a t the compact LIE groups K and L have the same rank; thus eve ry e lement of K o is conjugate to an ele- men t of a maximal torus of L and consequent ly has a fixed point on K / L ; it follows t ha t F ~ K o ---- (1}. As K o ---- K ~ I0(N ), th is proves F ~ Io(N ) ~ {1}.

Suppose t ha t F :/: (1}. Then the curva ture conditions on N and the results of w 2 .8 show tha t N ----- P~(C) and F is a 2-element group (1, ag} where a is the i somet ry induced b y conjugat ion of C over R and g c Io(P~((~)) is induced by some element (also denoted g) of the ma t r ix group SUa(m -~ 1). (~g)2 ~ 1 ~ 1' implies t ha t ~g-lg is a scalar ma t r ix c I ~ SUh(m + 1), so g ---- c tg. As t(tg) ~ g we have c---- :t: 1, so tg = -t- g. Le t Z b e the

of all matr ices (A On) where A e U ( h ) , B ~ U ( m - ~ 1 - h) and group

with those condit ions, for we m a y assume K = K o ,~ o~K o. I f g = ~g, then a = ta and b = tb, and there are un i t a ry matr ices u and v with ua*u -~ I h

and vbtv ~- I,~+1_ n. I t follows t ha t (det. u)~(det, v) 2 = 1, so we m a y replace

u o r v b y a s c a l a r m u l t i p l e , i f nece s sa ry , a n d a s s u m e h : ( 0 0 v ) e Z . Now

hgth = I , and this gives ad( th -1) (~g) = t h - l~g th = t h - l ~ h - l h g t h = ~ h - l ~ h - l =

= th-X" th~ ~ a , so ag has a fixed point on K ( y ) c P~a(@). This being impossible, we have g = - - t g . Now a = - ta and b = - t b , so N = N ~ gives t h a t r = h/2 -~ s/4 and t ~ (m ~- 1)/2 = ((n/2) ~- 1)/2 are integers.

( 0 0/w) for every integer w > 0 , and then define Define Jw---- - - I w

(o ) ~ SU(2r) • SU(2t -- 2r) c SUh(m ~- 1) . J _-- j~_~

Now we have un i t a ry matr ices u and v such t h a t u a~u ---- J~ and v btv ---- J t - r . As be-

I so t rop ic man i fo lds of indef ini te me t r i c 49

fore we may assume h-~- ( 0 ~ ) ~ Z . Now F is conjugate to (1, ad(th-i)(~g)},

and the second element is given by

th-1 o~gth ~ th-1 ~xh-1 hgth = th-1 ~ h - i J ~ th-1. th~ J _~ ~ J .

As a J has no fixed point on P~(C), the Theorem follows. Q . E . D .

6.4. A finiteness criterion. The latter part of w 6.2 consisted era technique which could be useful in a variety of situations. For this reason we state it separately.

Theorem. Let G be a connected semisimple LzB group with finite center, let H be a closed subgroup o/ G, and let F be a subgroup o/ G which acts properly discontinuously on G/H. Suppose that ~ is preserved by a C~RT~N involution a o/if), that ffJ -= R + ~ is the decomposition under a, and that ~ ~, ~ contains a CAnT~ subalgebra of the symmetric pair ((~, ~) . Then F is finite.

Corollary. Let G be a group F~, Spl(n), SUl(n) or S01(n). Let H be a closed noncompact subgroup o/ G whose LIB algebra is preserved by a C ~ T ~ involution of ~ , and let F be a subgroup of G which acts properly discontinuously on G/H. Then I ~ is finite.

For if G is semisimple, then, in the notation of Theorem 6.4, (ffi, R) has rank 1 and .~ ~ ~ # 0. I f G is not semisimple, then n is so small that G ~ H .

Remark. Par t 1 of Theorem 6.1 could be proved from Theorem 6.4; the proof then would be more uniform and would avoid appeal to [18], but the method used in w 6.2 is more elementary.

7. The classification theorem for isotropic manifolds

Extending the results ([17], w167 16-17) for constant curvature, we will prove:

7.1. Theorem. Let M~ be a connected locally isotropic pseudo-RzgM,4~ian mani/old which is not/lat. Then these are equivalent:

1. M'~ is isotropie. 2. M'~ is symmetric. 3. Let F be the group o/ deck trans/ormations o/the universal pseudo-RiRM~ian

covering xt : N~ ---> M'~. Then M~ is complete and F is a normal subgroup el I(N~).

4. There is a psendo-R~BM~NNian covering M ~ - ~ I ~ k ( R , C , K or Cay) or M~-~ H~'(R, C, K or Cay).

5. Either M~ is isometric to a model space P~' (C, K or Cay) or H~' (C, K or Cay), or there is a pseudo-REM~vian covering M~-+ P~ (R) or H~ (R).

4 CMH vol. 39

50 JOSEPH A. WOLF

Corollary. Let M'~ be a connected isotropic pseudo-RxBM,t~ian manifold. I] M~ is o/constant nonzero curvature, then it is explicitly described in ([17], Theorem 16.1). Otherwise, M~ is simply connected and is isometric to

R~, l~k(C, K or Cay) or H~'(C,K or Cay).

7. 2. Proof of Theorem. I f M~ is symmetric or isotropic, then it is complete. In the notation of statement (3.), if M~ is symmetric, then every symmetry lifts to a symmetry of N~, and consequently every symmetry of N~ is a ~t-fibre map. Thus every symmetry of N~ normalizes F . Now I 0 (N~) lies in the closure of the group generated by the symmetries and thus normalizes _1', and thus cent ra l izes / ' because /" is discrete. Now F lies in the centralizer Z of I o (N~) in I(N~). Given g EI(N~) and z E Z, one can check the various cases and see that gzg -1 is z or z-l; in fact this is trivial in nonconstant curvature and is contained in ([17], w 16) for constant curvature. Thus F is normal in I(N~). On the other hand, if we assume M~ to be isotropic instead of symmetric, we see that every isotropy subgroup of Io(N~) consists of ~r-fibre maps, and it follows as above t h a t / ' is normal in I (N~). Thus (1) implies (3) and (2) implies (3).

Assume (3). I f M~ is of constant curvature, then we know ([17], Th. 16.1) that (5) follows. I f M~ is of nonconstant curvature, then I0(N~) is centerless, and it is easily checked that the centralizer Z of I0(N~) in I(N~) is trivial; as F c Z, it follows that g is an isometry. Thus (3) implies (5). And it is clear that (5) implies (4).

Assume (4) and let ~ : M~-+ D~ be the covering of (4). Given d ~ D~, 9 ~I(D~) such that g(d) -~ d, and x ~ v/-l(d), there is a lift ~ r of g such that ~(x) = x. Now both (1) and (2) follow.

This completes the proof of Theorem 7.1. I f M~ is not flat, then Corollary 7.1 is the statement that (1) implies (5) in

Theorem 7.1; i fM~ is flat, it is the statement that (1) implies (3) in Theorem 15 of [17]. The Corollary follows.

Chapter II

Homogeneous Locally Isotropic Pseudo-RIEMANNian Manifolds

The goal of this Chapter is the global classification of complete homogeneous locally isotropic pseudo-RrsMAN~cian manifolds which are not flat. That classification was accomplished in an earlier paper [17] in the case of constant curvature. Our main technique is to reduce to the case of constant curvature and use the results of [17]. After the classification, we give some examples which show that the hypothesis of completeness is essential.


8. Application of the classification in constant curvature

8. 1. Theorem. Let M~ be a complete connected homogeneous locally isotropic pseudo-RzBM~Nian mani/old o] nonconstant sectional curvature. Let F be the group o] deck traus/armations o/ the universal pseudo-RIEM~N~ian coveriny ~: N;-+ M~. Then / '~Io(N~) = {1}.

Remark. I f N~ is not a complex elliptic or hyperbolic space, it immediately follows from Theorems 6.1 and 8.1 that M~ is simply connected and g is a global isometry; if M~ is not simply connected, it follows that M," is isometric to one of the two manifolds of (2c) and (2d) of Theorem 6.1. This does not complete the classification, however, as those manifolds must be checked for homogeneity.

Proo/. The possibility of reversing the metrics of M~ and N~ shows that we need only consider the case of non-negative involutive sectional curvature. If N~-~P~(Cay) with k < 2 , then Theorem 6.1proves F = (1}. I f N~---- = P~(Cay), then we reverse metrics and have several theorems (see [20], [21] or [11], for example) which imply triviality o f / ' . Thus we need only consider the case N~-~P~(F) with F = C or F~- -K.

Let A ~ F ~ I o (N~). According to Theorem 6.1, we need only prove tha t z] is finite. Thus we need only prove A finite under the assumption hr~ : P~'(F : C or K).

8.2. Let / be the degree of F over R, and define h and n by h : 8 : / k and n~- 1 = / ( m + 1 ) = r + f . We have ~0: S~-->P~(F) given by

Uk(m ~- 1 ; F)/Uk(m, F) --~ Vk(m -~- 1 ; F)/Uk(m, F) • U(1, F)

and a v/-equivariant epimorphism ~ : H--~ Io(P~(F)) where H = SU k (m ~ 1) or Sp k(m ~- 1). Define D : ~-I(A).

l~k(F ) consists of all positive definite lines in ~ + 1 , and

where the norm is taken with respect to the hermitian form on ~ + 1 . View 1~ +1 as a real veetorspace V ; F acts on V. Let Q be the real part of the hermitian form on F~k +1 ; Q is a symmetric bilinear form giving V the structure of R~ +1 . Let F' denote the multiplicative group of unimodular element of F ; now F' c Oh(n ~ 1) and H c 0a(n -t- 1) where everything acts on V and H is the group mapped by ~ .

8. 3. Lemma. Let G' be the centralizer of D in H and de/ine G-~ (7'. F'. Then G is transitive on the points o/ S~. In particular, S~/D is a homogeneous pseudo-R1~M~ian mani/old o/constant positive curvature.

52 JOSEPH A. WOLF

Proo~ o] Lemma. Let x, y ~ S~. Homogenei ty of M~ implies ([17], Th. 2.5) t h a t the centralizer of A in I 0 (N~) is t ransit ive on N~; it follows tha t G' is t ransit ive on N~; thus G' has an element gl mapping x F onto yF. As ]l xll s -~ ~--- ]l Y]I 2, F' has an element g2 sending glx to y . This proves t rans i t iv i ty of G.

Transi t iv i ty of G shows t h a t D acts freely on S~, for every element of G commutes with every element of D. The action of A on P~(F) is properly discontinuous and Ker. ~ is finite; thus the action of D on S~ is properly discontinuous. Now S~-~ S~/D is a normal covering, and homogenei ty follows ([17], Th. 2.5).

L e m m a 8.3 is now proved.

8. 4. Lemma. I f D is not [inite, then the R-linear action o ] D on V is not ~ully reducible.

Proo[ o] Lemma. L e m m a 8.4 is based on ([17], L e m m a 8.4). Suppose t ha t D is infinite and fully reducible. Then L e m m a 8.3 and ([17], Th. 10.1) show t h a t V is the direct sum U �9 W of two to ta l ly Q-isotropic R-subspaces and tha t there is a nonzero real number t ~= =t= 1 such tha t D has an element d whose restriction to U is t I , whose restriction to W is t - l I , and such tha t d and pos- sibly also - I generate D. U and W are F-subspaces of V because D is F- linear and t ~= t -1 . Le t QF be the F-hermit ian form on V = l~k +I ; m + 1

2k, k is the common dimension of U and W over F, U and W are tota l ly QF-isotropic ([17], w 8.4), and there are F-bases {et) of U and {/~} of W with Q~(et, ]j) : 2O~j. Now let ~ represent GL(k, F), the general linear group of

F k, on V by: ~ ( ~ ) h a s matr ix I o t a _ l ) in the basis {e , , /~}of V o v e r F .

The image of ~ is precisely our group G', consisting of all F-linear transformations of l z which preserve QF and commute with each element of D.

We have k > 1 because M~ is no t of constant curvature. Define w~ : (1/2) (e~ -- ],) and w~+~ = (1/2) ( e / + / , ) for 1 < i < k; now {%} is a QF-orthonormal basis of V. As w~+~ ~ S~ and G ---- G ' . F ' is t ransit ive on S~, whenever we have x r S~ we must have b r F' and ~ E GL (k, F) such tha t a(~) : wk+~-+xb. Let x -= 2:w~xj with xj �9 F ; then i t follows as in the proof

k k

of ([17], L e m m a 8.4) t ha t X(x~b) (x~+~b) r R c F , i.e., t ha t ,Y,x~xk+~ r R . i = l k i=1

As F ~ R, S~ has many elements x ---- X%x~ for which X x~x~,~ is not real. Thus D cannot be both infinite and fully reducible. ~=~

L e m m a S. 4 is now proved.

8 .5 . Lemma. I f D is not finite, then F # C.

P r o o / o / L e m m a . Suppose t h a t D is infinite and F = (3. Nonconstancy of sectional curvature of M~ shows m -f- 1 > 2, so the kernel Ker. ~ , which

I so t rop ic man i fo lds of indef in i te met r ic 5 3

is i somorphic to cyclic group Z,~+l, is a cyclic subgroup of D of order q > 2. D has an e lement d of infinite order; let D~ be the subgroup genera ted b y d and K e r . r Ker .~b is centra l in D s o D I ~ Z • Zq, q > 2 . L e m m a s 8 .3 and 8 .4 hold for D 1 . Now D 1 is abel ian bu t no t ful ly reducible on V ; T h e o r e m 10 . 1 o f [17] says t h a t D~ is i somorphic to Z , Z • Z 2, Z • Z or Z • Z • Z2. Thus F =/= t3 if D is infinite.

L e m m a 8 .5 is now proved.

8.6. The following is the last and mos t delicate s tep in the p roof of Theorem 8.1.

L e m m a . I [ D is not finite, then F ve K .

Proof of L e m m a . Suppose t h a t D is infinite and F ----- K . D is no t ful ly reducible on V and we a p p l y the m e t h o d of ([17], w 10.4). L e m m a s 8 .2 and 8.6 of [17] p rov ide a G- invar ian t D- inva r i an t m a x i m a l to ta l ly Q-isotropie subspace U of V and a Q-or thonormal R-bas is ~ ---- { v l , . . . , v,+~=h+~} such tha t (e i = v h + ~q--v i and [ ~ = v h + i - v i for 1 < i < p ) {ei} is a basis of U and {v~+ 1 . . . . . v n ; e a . . . . , %} is a basis of U • . Le t fl be the basis {[1, �9 �9 �9 f~ ; v~+ 1 . . . . . v h ; e a . . . . . %} of V and let B be the a lgebra of l inear t r ans fo rmat ions of V genera ted b y D . I f a is an e lement of B, D or G, then a preserves bo th U and U • so a has m a t r i x relat ive to fl given in block fo rm b y (: a) a n a~

a ~ a 4 a 5 .

0 a 6

t hen a c O h(n -k 1) and it follows t h a t a e = tal-1 and I f a e G or a e D ,

a, e-O(h - p) . Le t r l , ?'4 and r 6 be the ma t r i c represen ta t ions of respect ive degrees p , h - p

and p of G given b y a ->a~ , a - + a 4 and _a -->a6. r6 is i rreducible ([17], L e m m a 8.2) and r 1 is cont ragred ien t to r e. I f a e G sends vh+ 1 to x = X x j v j , t hen one checks t h a t the first row of a~ is (xh+ 1 - x l , . . . , xh+~, -- x~) = q ( x ) .

As x ranges over S~, q (x) ranges over a subset of l ~ on which no nondegenera te quadrat ic fo rm is bounded. I n [17] this was observed to imp ly t h a t r 1 has no nonzero s y m m e t r i c bi l inear invar iant . As G ---- G ' . K ' and K ' is compac t , it also shows t h a t the res t r ic t ion r~ of rl to G / is w i thou t a nonzero symmetric bi l inear invar ian t . I t follows t h a t r E = re I a, has no nonzero s y m m e t r i c bilinear invar ian t .

After considering bi l inear invar ian t s of rs, w 10.4 of [17] goes on to show tha t every e lement d �9 D has m a t r i x of the fo rm

d = d4 re la t ive to ~.

o dd

5 4 JOSEPH A. WOLF

I t is further shown that an element d of D can be chosen with da ~ 0 and that, this choice made, d3 is a bilinear invariant of r e, i.e., ~ged3ge -~ da for every g f G . Now let gEG ~ and u E K ~. Then tg6(d3ue) g e : ~ g 6 d 3 g 6 u s : d 3 u 6 , whence d3 ue is a bilinear invariant of r~. I t follows that d8 ue is antisymmetric for every u ~ K ~. Now as in [17] we take u, v ~ K ~, observe that w e K ~ gives d3w e ~ - ~(d3ws) -~ - ~w 6. tda -~ tw6d3, and conclude that uev6 -----

-~ (uv)6 ~ d3 -1" ~(uv)e " d3 ~ d3 -1" tv~" d3 " d3 -~" *u~. d3 -~ veue. As the restriction of r e to K' is faithful, it follows that K' is commutative. That is absurd.

Lemma 8.6 is now proved.

8. 7. We saw in w 8.1 that, in order to prove Theorem 8.1, it was sufficient to assume N~ --~ I~ (F ~ C or K) and prove A finite. For this, it is sufficient to prove D finite. I f D is infinite, then Lemma 8.5 shows F :/: C and Lemma 8.6 shows F :~ K. This completes the proof of Theorem 8.1. Q . E . D .

9. The classification theorem for complete homogeneous locally isotropic manifolds of nonconstant curvature

The following theorem gives the classification mentioned above. Combined with Theorem 12 of [17], it gives the classification up to global isometry of the complete homogeneous locally isotropic manifolds which are not fiat. A partial classification is available [19] in the fiat case.

9. 1. Theorem. Let M'~ be a complete connected locally isotropic pseudo-Rx~-

MAt~Nian mani /o ld o~ nonconstant sectional curvature. T h e n these are equivalent:

1. M'~ is homogeneous.

2. ~ ( M ~ ) is f inite. 3. ~ ( M ~ ) has only 1 or 2 elements.

4. M~ is isometric to one o / t h e sTaces :

(a) P~(Cay) or H~(Cay)where n - ~ 16 and h -~ 8k. (b) l~ (K) or H~(K) where n - ~ 4 m and h : 4 k .

(c) P~(C) or H~(C) where n : 2 m and h - ~ 2 k .

(d) P~-I(C)/(1, ocJ~ where n -~ 4t - 2, h -~ 4s , o~ is the isometry induced

by conjugation o/ C over R , and J is the isometry induced by

(O )o - - 1 . 0 0 It-. ~SU~"(2t)"

~ I f _ B

(el o J} w h e r e = 4 t - 2 , h = 4 r - - a n d a n d J are

given as above wi th s = t - r .

Isotropic manifolds of indefinite metric 5 5

9. 2. Proo]. Theorem 6.1 is the equivalence of (2), (3) and (4), Theorems 8.1 and 6.1 show that (1) implies (4), and the manifolds (4a), (4b) and (4c) are homogeneous. As one passes from (4e) to (4d) by reversing the metric, the Theorem will now follow as soon as we prove tha t P2~, t-1 (C)/{1, ~J} is homogeneous.

Let V ~ C~, t and identify J with its matrix as given in (4d). Let A be the

algebra of R-linear endomorphisms of V generated by ~ J and V ~- 1 I~t. As

both a J and 1 / / - 1 I~t have square - I , and as they antieommute, A is a quaternion algebra. Let Qc be the C-hermitian form on V and let Q be its real part. There is a unique A-hermitian form Qa on V whose real part is Q. We now have ( V , Q ) = R 4 4 t , , ( V , Q e ) = C ~ , t and (V, Q a ) = K t. One can describe S4t-1 by Q ( x , x ) ~ - 1 by Q v ( x , x ) = 1 or by Qa(x x ) = 1. The unitary group Sp'(t) of (V, Qa) is a subgroup of the uni tary group V2'(2t) of(V, Qc) ; by construction, SpS(t) is the centralizer of a J in UeS(2t). As Spa(t) acts

.~4t-1 and C-linearly on V, it induces a transitive group of transitively on -4, motions of ~t-1 P2, (C). I t follows that 2t-1 P2s (C)/{1,aJ} has a transitive group of isometrics. Q . E . D .

9.3. Combining Theorem 9.1 with Theorem 12 of [17], we obtain the rather strange result:

Theorem. Let M~ be a complete connected locally isotropic pseudo-RxEM~ivNian manifold which is homogeneous but not isotropic. Then M~ is of conetant sectional curvature, if and only i[ ~1 (M~) is not of order 2. M~ is of nonconetant sectional curvature, i/ and only if n ~ 4t - 2, h ~ 4s (resp. h : 4r - 2)

p2t--1 H~_ 1 (C)/{1, ~J}) . and M~ is isometric to ~ , (C)/{1, ~J} (rezp. 2t-i For if N~ is complete, connected, of constant sectional curvature, and with

fundamental group of order 2, then N~ is isometric to P~(R) with h < n - 1 or to H~(R) with h:> 1. Those manifolds are isotropic. Theorem 9.3 now follows from Theorems 7.1 and 9.1.

10. A characterization and completeness criterion for locally symmetric affine homogeneous spaces

This section provides a tool for the s tudy of incomplete homogeneous spaces.

10. 1. An a/fine mani/old is a differentiable manifold with an affine connection on its tangentbundle. I f M is an affine manifold, then A (M) denotes the group of all connection preserving diffeomorphisms of M, and M is a/fine homogeneous if A (M) is transitive on the points of M. Given x ~ M, there is an open neighborhood U of x on which the geodesic symmetry at x is a diffeomorphism; M is locally symmetric if, given x E M, U can be chosen such that

56 JOSEPH A. WOLF

the geodesic symmetry preserves the induced connection on U; this is equivalent to vanishing of the torsion tensor and parallelism of the curvature tensor for the affine connection on M. M is symmetric if it is locally symmetric and the geodesic symmetries extend to elements of A (M). I f the affine connection of M is the LEVI-CIVITA connection of a pseud0-RIEMANNian metric, then M is (locally) symmetric in the affine sense if and only if it is (locally) symmetric in the pseudo-RI~.MAN~ian sense, for an element of A (M) or A(U) is an isometry if it is an isometry at some fixed point; on the other hand, M can be affine homogeneous but not pseudo-RIEMANNian homogeneous.

10.2. Theorem. Let M be a locally symmetric connected a//ine homogenemt~ mani/old. Then there is a connected simply connected a//ine symmetric manifold N , a point y ~ N , and a closed connected subgroup B c A (N), such that

1. B (y) is an open submani/old of N . 2. M and B(y) have the same universal a/line covering mani/old. 3. M is complete i / a n d only i/ B (y) = N .

10.3. Corollary. Let M be a compact connected locally symmetric af/ine homogeneous mani/old. I / ~I(M) is/inite, then M is complete.

For B(y) must be closed in N, as it is a continuous image of the compact manifold which is the universal covering of M.

10.4. Corollary. Let M be a connected locally symmetric a//ine mani/old, and suppose that there is a connected solvable subgroup G c A(M) which is transitive on M . I] the space N o/Theorem 10.2 is not contractible, then M is not complete.

For the proof of Theorem 10.2 will show that B can be chosen to'be solvable when A (M) has a solvable transitive subgroup. I f M is complete, then B (y) ----

N, so N -~ B / F where F is the isotropy subgroup of B at y. F is closed in B, and is connected because n~(N) = (1}; it follows tha t N has a defor- mation retraction onto a torus. As 7~a(N) = (1}, this says that N is contractible. Thus M cannot be complete if N is not contractible.

10. 5. Proof of Theorem. Choose a connected transitive subgroup G c A (M), close it i f it is not closed, let x ~ M, and let H be the isotropy subgroup of (7 at x. The universal covering fl: G'-+ G gives an action of the universal covering group G r on M by g ' :m-+f l (g ~) (m). A result of K. No~Izv ([15], Th. 17.1) provides a connected simply connected affine symmetric manifold N, a point y c N, open neighborhoods U of x and V of y, and an affine diffeomorphism / of U onto V carrying x to y. Let W be an open neighborhood of I ~ G ' such tha t g'(x) EU for every g '~W. Given g ~ W , now, we have a small open neighborhood X of x such that X (J gr (X) c U; thus g' induces an affine equivalence g" of ] (X) onto /(g' (X)) .

Isotropie manifolds of indefinite metric 57

N is complete because one can ex tend geodesics b y the symmetr ies . As N is s imply connected, and as its affine connection is real analyt ic ([15], Th. 15.3), it follows t ha t g" extends to a unique e lement (also denoted g") of A (N). This map g' ~ g" gives a homomorph ism of G' onto a subgroup L of A(N) . L(y) contains V and thus is an open submanifold of h r . Le t B be the closure of L in A(N) . B preserves the closed set N - L(y) because so does L , and thus B(y) = L(y) . The first s t a t emen t of the Theorem is proved. For Corollary 10.4, one observes tha t B is solvable if G is solvable.

In the universal covering fl : G' --> G, let H ' be the iden t i ty component of f l- l (H), fl induces a universal covering ze:G'/H'--~ M; pull back the connection of M so tha t ~ is the universal affine covering. Now let ~x : G' --* B(y) be defined by ~(g') = 9"(y). o~(H') = y because every veetorf ield on U corresponding to a 1-parameter subgroup of H vanishes a t x. Thus ~ induces a map �9 of G'/H' onto B (y), and ~ is a covering. I f we take a small ~z-admissible open set in M , lift to G'/H' and then map b y ~ , we have an affine equivalence, for everyth ing can be t rans la ted back into U and V ; thus r is an affine covering. The second s t a t emen t of the Theorem is proved.

The second s t a t ement shows t h a t M is complete if and only if B(y) is complete. As hr is complete, M is complete if B(y) = _IV. Now suppose B(y) :/: N ; we mus t prove t ha t B (y) is not complete. As B(y) :~ N bu t B(y) is open in hr, we have a bounda ry point v of B(y) with v ~ B(y) . Some geodesic of N through v must have a point in B(y); otherwise v would have a geodesic coordinate neighborhood disjoint f rom B (y) and thus would no t be a b o u n d a r y point. Le t {vt} be a geodesic of N , v 0 = v and v I E B(y) . vl+ . ~ B(y) for I~1 small; thus B(y) contains an arc of (vt}. I f B(y) were complete it would contain all of {vt}. As v ~ B(y ) , B(y) is no t complete. Q . E . D .

10. 6. The following can occasionally be useful in applying Theorem 10.2.

Theorem. Let N be a complete connected a/line homogeneous mani/old; let B c A (N) be a closed connected subgroup with an open orbit B (y); let H be the isotropy subgroup o/ the identity component Ao(hr); and let KB, K H and K~ be respective maximal compact subgroups o/ B , H and Ao(N ) such that K~ c K~ and K H c K a . I / K B is not transitive on K~/K H, then B(y):/:IV, and consequently B (y) is not complete.

For we can assume L = K~ n KB to be maximal compact in H N B . I f K B is not t ransi t ive on K~/KB, then an orbi t KB/L has lower dimension and thus (these manifolds are compact) has different h o m o t o p y type . Then B(y) has different h o m o t o p y type than hr, for B(y) = B / (B N H) retracts onto KB/L and N = Ao(_hT)/H re t rac ts on to K~/K~. I t follows t h a t B(y) ~ N .

5 8 JOSEPH A. WOLF

11. k tool Ior the construct ion oI incomplete manifolds

We will develop tools for the construct ion of incomplete homogeneous pseudo-RI~MANNian manifolds as orbits of groups which preserve the exponential image of a set obta ined f rom a light cone. The basic idea is i l lustrated by :

l l . 1. Theorem. Let U be a nonzero totally isotropic linear subspace o / R E, and define M~ to be the open set R'~ -- U -L with the induced pseudo-RxEM~NNian structure. Then M~ is a flat homogeneous incomplete pseudo-Rz~MaNivian manifold.

Remark. There are min. {h, n - h} non-isometr ic manifolds M~ above, corresponding to the possibilities for the dimension of U.

Remark. Given a group H of l inear t ransformat ions of U which is t ransi t ive on U - {0}, define G to be the subgroup of I(R~) consisting of elements which preserve M~, whose homogeneous par t preserves U, and whose homogeneous par t restr icts to an element of H . We will in fact p rove t h a t G is t rans i t ive on M~.

Proof. Let fl be a skew basis o f R~ wi th respect to U. This means tha t we choose an o r thonormal basis { v l , . . . , vh; vh+l . . . . , vn} of R~ such tha t (t = dim. U; /i : - vi + vn-t+i and e~ = v~ + v~_t+ ~ for 1 < i < t ) {e~, . . . ,et} is a basis of U, and we define fl to be the basis {fl . . . . f~ ; vt+ 1 . . . . . v~_t ; el, �9 �9 �9 et} of R~. U • is spanned b y the e~ and the vj in fl ; thus G ~ = ( g e 0 h (n) : g (U) = U} consists of the elements g ~ 0 ~ (n) whose ma t r ix relat ive to fl has block form

g~ ----- g4 g5 �9

0 g8

L e t H be a subgroup of GL(U) which is t rans i t ive on U - {0}. There is a monomorph i sm r ~ r' of H onto a subgroup H' of G given by : ,r0 !)

(r')~ = i : I 0

where r denotes the ma t r i x of r wi th respect to (e~}. Le t G be the subgroup of I (R] ) consisting of all isometries whose homogeneous pa r t lies in H ' and whose t rans la t ion par t is in U ~ ; G preserves M~. Given x c M~, x = / § v where v c U" and / is a nonzero e lement of the span of the /~ ; choose r ' ~H' wi th r ' (11) = ] ; G contains the t rans format ion y -+ r' (y) + v, and this t rans- fo rmat ion sends/1 to x . Thus G is t rans i t ive on M ] . Q . E . D .

Corollary. The connected noncommutative 2-dimensional LIE group admits an incomplete flat left-invariant Lo~B~Tz metric.

I s o t r o p i c m a n i f o l d s of i n d e f i n i t e m e t r i c 5 9

Proof. Suppose n = 2 and t = h = 1 above. Then H consists of nonzero scalars and G is simply t ransi t ive on M~. The diffeomorphism g - , g (fl) of G onto M~I endows G with an incomplete flat lef t - invariant LORENTZ metric, and the group of the Corollary is isomorphic to the iden t i ty component of G. Q . E . D .

11.2. In order to app ly the idea of Theorem 11. 1 to a manifold which is symmetric bu t no t flat, one replaces H ' b y a corresponding subgroup of the isotropy group at a point, and one replaces t ranslat ions along U" by t ransvec- tions. I t is difficult to see tha t a group analogous to G will then result, and it is ve ry difficult to see whether the orbits are proper submanifolds. Bu t the construction of most of our model spaces allows us to replace those considera- tions of LIE algebra by considerat ions of linear associative algebra while preserving the idea of Theorem 11.1. The associative algebra consists of:

11.3. Theorem. Let QF be the hermitian /orm on F'~, define quadrics S = {x ~ F~k:QF(x, x) -= 1~ and H -~ (x ~ ~ : Qp(x , x) = - 1~, and let U be a nonzero maximal totally QF-isotropic F-subspace of F~k . Let P be any group of F-linear transformations of U which contains the real scalars and is transitive on U - (0~, define G-= ( g E U k ( m , F ) : g ( U ) - = U and g l v E P } , let F' be the group of transformations of F~ which are scalar multiplication by unimodular elements of F, and let G' be the group o /R- l inear transformations o / ~ generated by G and F'. Let oc be a QF-orthonormal F-basis (v 1 . . . . . v~; v~+ 1 . . . . . v,,} o/ F~k which gives a skew F-basis fl of F~k with respect to U. I[ 2]c > m, then G t (vl) is a proper open submanifold of H. I[ 2 ]c < m, then G' (v,~) is a proper open submanifold of S.

Proof. Le t t = dim.FU and f = dim.RF so t / = d im.aU, let fl--~ (/1 . . . . . f t ; v~+~ . . . . , v,~_ t ; e I . . . . , e~} be the skew F-basis of F~k der ived f rom ~ so tha t

t

e -~ (el . . . . . e,} spans U over F, and let S ~t-1 be the uni t sphere (Ze~a~ : g t 1

Zany-----1} in U. P acts t rans i t ive ly on S ~1-1 by p : u - ~ p ( u ) . ( 2 : ~ b ~ } -1/~ 1 1

where u ~ S ~1-1 and p(u) = Xe~b i ; i t follows t ha t a maximal compact subgroup K = P is a l ready t ransi t ive on S ~1-1, by a check i f tf ~ 2, and because S ~I-1 is simply connec ted if t / :> 2. We m a y now al ter e, changing fl and ~ in consequence, so tha t the F-linear act ion of K preserves S'I-~; this change will not effect the Theorem.

We wish to p rove certain orbits G'(v) proper and open in quadrics Q, i.e., properly conta ined in Q and of the same dimension as Q. We m a y cut G ~ to a subgroup in proving G' (v) open, and we m a y do the same in proving G' (v) r Q because our reason for t ha t inequal i ty (see w 11.5) will depend only on m, k and the fac t t ha t G ~ ( U ) = U. Thus we m a y replace P b y its subgroup

60 JOSEPH A. WoLF

genera ted by K and the real scalars. This done, the ma t r ix p, of an e lement p e P relat ive to esat isf ies c . i % : t ~ - i where 0 : / : c E R .

Le t g ~ G. I t s ma t r ix re la t ive to fl has block form

g/~ = g4 (/5 ,

0 (/e

and g6 is a real scalar multiple of 91 because gs = P. for some p ( P by defin- i t ion of G, gx ---- tg6-1 because g e U k (m, F), and t ~ - i is a real mult iple of p, . Le t (gi)u and (g~)~, denote the u row and the (u, v) e lement ofg~. Then we have

t m- - t

g(el -4-/1) : X {e,(g.)l, A- f,(gl)l, -4- ei(g3)ii} -~ ~ Vi(g2)li 1 t + l t m - t

g (e l - - /1) : ~ {ei(g6)li - - / i (g l )1 i - - ei (g3)1t} - - ~ v i (g2)1t 1 t-i-1

thus 2v 1 --~ e I -- /1 tells us t h a t 9(vl) ----- v 1 if and only if (a) (g~)l = ( 0 , . . . , 0) ( b ) ( g l ) l = ( ] , 0 , . . . , 0 ) = (g6) l - - (g3)l �9

Similarly 2vm_t+ 1 -~ el -4-/1 says t ha t 9(v~_t+l) = v~_t+ 1 i f and only if ( a t ) (g2)1 = ( 0 , . . . , 0 )

(b') ((71)1 = ( 1 , 0 . . . . . 0) = (gs)~ -f- (g3)1. Now (a) coincides with (a'), and the first equa l i ty of bo th (b) and (b') imply (gl)l = (g6)1, so (b) is equivalent to (b'). I t follows t h a t y(vl) = v I if and only i f g(v,n_t+l) = v,n_t+ 1. In part icular , as vm E G(v~_t+l) so tha t G(v,~_t+l) = = G(v, ,) we have dim. G(vl) = d im . G(v , , ) .

To calculate these dimensions we need:

11 .4 . Lemma. Reta in the notation o / T h e o r e m 11.3. I / 2 k >_ m , then G is transi t ive on S . I / 2 k <_ m , then G is transi t ive on I I .

Replacement of QF by - QF shows the two s ta tements to be equivalent , so only the first need be proved. Assume 2k > m; then t ---- m - k because U is m a x i m a l to ta l ly isotropic, whence the span of {v~+ 1 . . . . , v,~_t_k} is negat ive definite under QF. One now repeats the t rans i t iv i ty a rgumen t of ([17], p. 127) replacing EucLIDean pairings b y F-hermit ian pairings and writ ing scalars on the right.

11 .5 . Combining L e m m a 11.4 with dim. G(vl) --= dim. G(v, , ) , we see that G(vl) and G'(vl ) are open submanifolds of tI , and t h a t G(vm) and G'(v,,)

are open submanifolds of S. As G and G' preserve U, we know ([17], Lemma 8.2) t ha t t r ans i t iv i ty of G or G' on S would imply 2k >_ m, and thus, by reversal of QF, t r ans i t iv i ty of G or G' on II would imply 2k < m. The Theorem follows. Q . E . D .


11.6. A conjecture. Theorem 11.3 will allow us to deal with all of the isotropic manifolds of str ict ly indefinite metric except P~ (Cay) and H~(Cay). In those spaces, however, we will see t ha t the normalizer of a BORF.L subgroup of the full group of isometrics has an open proper orbit. I believe tha t this is the case for every symmetr ic pseudo-RIEMA~Nian manifold G/H of str ict ly indefinite metric and of rank one where G is simple and has the same rank as H. Unfor tunate ly I have been unable to give a general proof of this conjecture, even for the spaces covered by Theorem 11.3.

11. 7. Theorem. Let B be a BOlCgL subgroup o] F* 4 and let P be the normalizer o~ B in F*. Then P has a proper open orbit on F*4/Spin(9).

Proo/. Let G = F* acting on M = F*/Spinl(9). Choose x E M , let H be the isotropy subgroup of G at x, and choose a maximal compact subgroup K of G which is normalized by the symmet ry s of M at x. We have involutive automorphisms ~ and a = a d ( s ) of ffi such tha t ffi---- ~ A - ~ = . ~ A - ~ I ~ where a is -~ 1 on ~ and is - 1 on ![R, and where v is + 1 on ~ and is - 1 on ~ .

Metrize M as tI~ (Cay) with the KILLIXG form of ffi ; let l[ be an isotropic (= light like) line in M r : 2/~, let ? be the light like geodesic exp (lI) �9 x in M, and choose y r ~, within a geodesically convex normal coordinate neighborhood of x. The point y cannot be fixed under exp (s for exp (s generates G. Thus we have an one-dimensional subspace ~I c s such tha t 9~ ~ .~', where H ' is the isotropy subgroup of G at y .

A = exp(9~) is maximal among the algebraic tori of G which split over R, for the RIEMAN~Cian symmetr ic space G/K is of rank 1; thus there is a unipotent subgroup N c G such tha t G = K A N is an Iwasawa decomposition. We may replace B by a conjugate, assuming B -= A N . Now P = Z B where Z is the centralizer of A in G. ~(3) = 3 because ~(9~) = 2 ; thus 3 = ~A-~R where ~ = 3 fl ~ and ~ R = 3 f3 ~ . ~ - = 9 ~ as rank ( G / K ) = 1, and Q = exp (~ ) is compact; P = Q A N .

Let J = P n H ' . J is algebraic because P and H ' are algebraic; thus 3 = ~ A- E -f- ~ where !3 is a unipotent ideal, ~ § E is reductive, ~ is semisimple, and E is the center of ~ § ~ . S = exp (~) is a compact subgroup of H which preserves the geodesic ~: compactness follows because S is a semisimple subgroup of the compact group Q, and ? is preserved because S fixes both x and y . In proving H~((3ay) to be isotropic, we saw t h a t the maxima among the compact subgroups of H preserving a nonzero element of the light cone of M , were of type G~. I t follows tha t S is contained in a subgroup (~2 of H ; in particular, dim. ~ _< 14. E § !3 lies in a BOREL subalgebra ~ ' of ,~ ' . !D' is of dimension S because H' / (maximalcompact) = Spin 1 (9)/Spin(S) is real hyperbolic 8-space; thus dim. (E + !8) _< 8. I f dim. (E -4- !l~) = 8,

62 Jos~rH A. WOLF

then ~ ~ ~ = ~B'; it would follow tha t ~ + !~ contains the LIE algebra of an algebraic torus split over R; ~[ being the only such algebra in ~ because Q is compact and ~ is the only such algebra in ~B, it would follow tha t 9~ c ~ ' ; by our choice of ~ , this is not the case. Thus dim. ((~ + ~) < 7. This proves dim. 3 < 2 1 .

= ~ § ~ direct sum as P = QAN. !~ is of dimension 16, 16 being the dimension of ]~(Cay) = G/K. Q is the centralizer of ~I in K, which is isomorphic to Spin (7). Thus !15 has dimension 37. I t follows that dim. P (y) = = dim. P-dim. J > 16, which proves that P(y) is open in M. On the other hand if P were transitive on M, then we could retract M onto the orbit of a maximal compact subgroup K' of P , and it would follow tha t K' is transitive on a sphere S s ---- K/(K f3 H) ; then K' would have to be of type B4 and would thus be a maximal subgroup of G; as P is not compact it would follow that P = G, which is impossible because B is normal in P and G is simple. We have proved tha t P (y) is a proper open subset of M. Q . E . D .

12. Existence of incomplete homogeneous spaces

The following shows that the hypothesis of completeness is essential in Theorems 9.1 and 9.3, in Theorem 12 of [17], and in Theorems 1 and 2 of [19].

12.1. Theorem. Let N~ be an isotropic pseudo-Rx~MA~Nian mani/old. Suppose either that 1V~ is o/ non-negative involutive sectional curvature with

n 0 < h ~_ ~ or that N~ is o/ non-positive involutive sectional curvature with

n < h < n. Then there is a subgroup G c I (N~) and a point y E N~ such that 2 - - G (y) is a proper open subset o/N~ ; in the induced pseudo-RiEM~ian structure, G(y) is an incomplete homogeneous pseudo-RzBM~ian mani/old locally isometric to IVy.

Remark. Let M~ be a non-fiat manifold G(y) constructed in the proof of Theorem 12.1. The construction of G and the arguments of ([17], w 10) imply: I] F is a properly discontinuous group o/isometrics acting freely on M~, and i] M~/F is homogeneous, then F is/inite. The signatures of metric involved arc about the same as those involved in the finiteness statement (part 1) of Theorem 6.1. This suggests tha t Theorem 6.1 (part 1) might be valid without assuming completeness.

12.2. Theorem 12.1 is contained in Theorem 11.1 if N~ is fiat, and is con ~ tained in Theorem 11.7 if N~ is a CAYLE:e plane. Reversing metrics if necessary,


we may assume N~ to be of non-negat ive involut ive sectional curvature . Now by Theorem 7.1 we have reduced to the case where there is a pseudo-RIEMANN- jan covering g : N ~ - + I ) ~ ( F ) wi th F : R, G or K .

n Our hypothesis 0 < h < ~ t ransla tes to 0 < 2s _< r .

Recall the quadric S : (x cF~+l: I]xlr--~ 1} where norm is in the her- Fr+1 F r mitian form on -s . Le t be the group of scalar mult ipl icat ions of F "+1 M S

by unimodular elements of F . As 0 < 2s < r q- 1, Theorem 11.3 yields a subgroup G' c U ' ( r q - 1 , F ) a n d a p o i n t b e S such t h a t ( H = G ' . F ' ) H ( b ) is a proper open subset of S. The project ion fl : S-+ S/F' ~ Psr(F) induces a homomorphism a : H--> H / F ' -= H " c I(P~(F)) ; H(b) = f l - l (H"( f lb ) ) because F / c H . Thus H"( f lb ) is a proper open subset of Psr(F).

Choose y e zr-l(flb) and lift H 1' to a group G of isometrics of N~; G(y) = = ~t- l (H"( f lb)) and so G(y) is a proper open subset of N~, necessarily incomplete in the induced metric. Q . E . D .

12.3. The group G' in the proof of Theorem 12.1 was constructed, in Theorem 11.3, from any subgroup P of the group of all F-hnear t ransformat ions of a maximal to ta l ly isotropic subspace U of F~ +x such tha t ( 1 ) P is t ransi t ive on U - {0} and (2) P contains the real scalars. Often there are several such groups, bu t I do no t know whether the result ing incomplete homogeneous spaces are distinct. Condit ion (1) cannot be avoided, bu t condit ion (2) was used only to ensure t ha t (3) P has a subgroup which satisfies (1) and whose i so t ropy subgroup a t any u c U - {0} preserves a subspace of U complemen ta ry to uF . Condition (2) implies t ha t P has no R-bihnear invar iant . Bu t i f a group P could be ibund satisfying (1) and (3), then the resulting group G' - F ' would have in its center one of the hyperbol ic t rans la t ion groups ~ (q-) described in ([17], w 9.2). I f this could be shown to act discontinuously on the resulting orbit M] ~: P~(F), t hen M'~/Ys would be homogeneous. This would be of interest in comparison to Theorems 6.1, 8.1, 9.1 and 9.3.

The Univers i t y o / C a l i / o r n i a at Berke ley

64 JOSEPH A. WOLF Isotropie manifolds of indefinite mean

R E F E R E N C E S

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[3] A. BOREL, Lectures on aymmetrie spaces, notes, Massachusetts Ins t i tu te of Technology, 1958. [4] A. BOREL and HARISH-CHANDRA, Arithmetic subgroups o/algebraic groups, Annals of Mathe-

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[11] S. KOBAYASHI, Homogeneous RI~MANN/an mani/olds o/negative curvature, Bull. Amer. Math. Soc., 68 (1962), 338-339.

[13J G.D.MoSTOW, On covariant fiberings o/ KLEIN spaces, American Journal of Mathematics, 77 (1955), 247-278.

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[16] H. SAMELSO~, Topology o/ LI~. groups, Bull. Amer. Math. Soc. 58 (1952), 2-37. [ 17] J .A. WOLF, Homogeneous mani/olds o/constant curvature, Commentarii Mathematici Helvetici,

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(1962), 77-80. [19] J.A.WoLF, Homogeneous mani/olds o/ zero curvature, Transactions of the American Mathe-

matical Society, 104 (1962), 462-469. [20] J .A.WoL~, Locally symmetric homogeneous spaces, Commentarii Mathematici I-Ielvetici, 37

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Journal of Mathematics, to appear.

R e c e i v e d J a n u a r y 14, 1963

isotropic manifolds of indefinite metric

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