homogeneous manifolds of constant curvature

Offprint of Commentarii Mathematici Helvetici, vol. 36, fasc. 2, 1961

Homogeneous Manifolds of Constant Curvature1)

by JOSEPH A. WOLF 2, Princeton (N. J.)

Chapter 1.

Chapter 11.

C h a p t e r I I I .

Contents Page

w 1. In t roduc t ion . . . . . . . . . . . . . . . . . . 113

The covering theorem for space forms

w 2. Pseudo-Riemannian manifolds . . . . . . . . . . 114 w 3. The model spaces R~ . . . . . . . . . . . . . . . 115 w 4. The model spaces S~h and H~ . . . . . . . . . . . 116 w 5. The universal covering theorem . . . . . . . . . . 118

Homogeneous manifolds of nonzero curvature .

w 6. The real division algebras . . . . . . . . . . . . 119 w 7. The 6 classical families of space forms . . . . . . . 120 w 8. Behavior of a quadric under field extension . . . . 121 w 9. The 8 exceptional families of space forms . . . . . . 126 w 10. The classification theorem for homogeneous manifolds

covered by quadrics . . . . . . . . . . . . . . . 130 w 11. The difficult s ignature . . . . . . . . . . . . . . 137 w 12. The classification theorem for homogeneous manifolds

of constant nonzero curva ture . . . . . . . . . . 140

lsot ropic and symmetr ic manifolds of constant curvature .

w 13. The not ion of a symmetr ic or isotropic manifold 140 w 14. The classification theorem for symmetr ic manifolds of

zero curva ture and for nonholonomic manifolds 141 w 15. The classification theorem for isotropic manifolds of

zero curva ture . . . . . . . . . . . . . . . . . 142 w 16. The classification theorem for symmetr ic and isotropic

manifolds of constant nonzero curva ture . . . . . 143 w 17. Applications . . . . . . . . . . . . . . . . . . 146

1) This work was begun at the Mathematisches Ins t i tu t der Universiti i t Bonn. Some of the results were presented at the International Colloquium o~ Di]]erential Geome2ry and Topology, Ziirieh, June, 1960.

I) The author thanks the National Science Foundation for fellowship support during the preparation of this paper.

H o m o g e n e o u s M a n i f o l d s o f Constant C u r v a t u r e 113

1. Introduction

In a recent note [6] we classified the l~IEMA~ian homogeneous manifolds of constant sectional curvature; this paper extends the techniques of that note to the classifications (up to global isometry) of the homogeneous pseudo- R i v . M ~ i a n (indefinite metric) manifolds of constant nonzero curvature, the symmetric pseudo-RIEMANNian manifolds of constant curvature, the isotropic pseudo-Ri~MAN~ian manifolds of constant curvature, the strongly isotropic pseudo-RIE~AN~ian manifolds, and the complete pseudo-RIEMA~ian manifolds in which parallel translation is independent of path.

ChapterI begins with a review of the necessary material on pseudo- RIE- mA~ian manifolds. Just as an n-dimensional RI~MANNian manifold M is denoted by M ~, so an n-dimensional pseudo-RI~A~cian manifold M,

h n

with metric everywhere of signature -- Z' x~ -~ 27 x~, is denoted by M~. 1 h + l

We then give euclidean space the structure of a pseudo-RIv.MAN~ian manifold h n.-.Bl

R~ of constant zero curvature, and give the quadrie -- 27 x~ ~- 27 x~ ~- 1 in 1 h+l

R ~+1 both the structure of a pseudo-RiE~A~Nian manifold $~ of constant positive curvature and that e r a manifold H~_ h of constant negative curvature. For 3~, this is essentially due to S. HELGASO~ (E2] and private communi-

cations). We then define $~ and / ~ to be the respective connected, simply connected manifolds corresponding to $~ and H~, and prove that a connected complete pseudo-Ri~mAN~ian manifold M~ of constant curvature admits

R~', 5~ or / ~ as universal metric covering manifold. Together with our criterion of homogeneity for manifolds covered by a homogeneous manifold (Theorem 2.5), this is the basis of our investigations.

Chapter I I is the classification of homogeneous pseudo-RIm~_A~ian manifolds of constant nonzero curvature. We define 14 families of homogeneous manifolds covered by $~ (w 7 and w 9), prove that every homogeneous manifold covered by $~ lies in one of those families (w 10), and then take care of the

signature h = n -- 1, where S~ is quite different fl'om S~ (w 11). The basic new technique is consideration of the behavior of a quadrie under field extension. We also rely on the elementary theory of associative algebras and on the version of SCH~'s Lemma which says that, if ~(V) is the algebra of linear endomorphisms of a vectorspace V over a field F, and ff G c E(V) acts irreducibly on V, then the centralizer of G in ~(//) is a division algebra over F.

Chapter I I I consists of classifications which are easy or which follow easily from Chapter II, and is best described by the table of contents.

114 JOSEPH A. WOLF

References are as follows: Lemma 8.3 refers to the lemma in w 8.3, Theorem 5 refers to the theorem in w 5, etc.

I wish to thank Professors S. HELGASOI~ and A. BOREL for several helpful discussions. HELGASOI~ suggested that I work on these classification problems; he proved Lemma 4.3 for h----- 1 and h = n - - 1 [2], and the proof given here is essentially the same; he communicated the proof of Lcmma 4.2, as well as another proof of Lemma 4.3, to me. BOWEL suggested that I look at the group [ (G/. (s, R)) of Lemma 8.4 to find a reducible linear subgroup of 0 s (2 s) transitive on the manifold S 2e-1 in R 2~.

Chapter I. The covering theorem for space forms

2. Pseudo-Riemannian manifolds

2.1. We recall a few basic definitions and theorems on pseudo-RI~MAN~ian manifolds, primarily in order to establish notation and terminology.

A pse~o-RIEMA~ian metric on a differentiable manifold M is a differentiable field Q of nonsingular real-valued symmetric bilinear forms Q~ on the tangentspaces M~ of M ; (M, Q) is then a pseudo-RIEMA_~ian mani/o[d. The pseudo-RI~MA~ian metric Q on M defines a unique linear connection, the L~.w-CI~TA connection on the tangentbundle of M, with torsion zero and such that parallel translation preserves the inner products Q~ on tangentspaces. (M, Q) is complete if its L]~w-CrwTA connection is complete, i. e., if geodesics can be extended to arbitrary values of the affine parameter.

2.2. I f X, Y c M~ span a nonsingular 2-plane S c M~ (Q~ is nondegenerate on S), then the sectional curvature of (M, Q) along S is defined to be 3)

K (S) = -- Q , ( ~ ( X , Y )X , Y)/ {Q~(X, X).Q~(Y, Y) - Q,(X, y)2)

where ~ is the curvature tensor of the LEvI-CMTA connection. (M, Q) is said to have consent curvature K if K ( S ) = K for every nonsingular 2-dimensional subspaee S of a tangentspace of M. Let x-* (xa , . . . , xn) be a local coordinate system valid in an open subset U of M; the coefficients of . ~ are given by

, and R~j~l=Xqj,,R~r%~.

s) This was brought to m y a t t e n t i o n by S. H_~Aso~r.

Homogoneous Manifolds of C ons t an t Cu rva tu r e 115

Then (M, Q) has constant curvature K if and only if ~)

R,~kz (p) = K (q,~ (p) q,, (p) -- q,~ (p) q~ (p) )

for every p E U and every (i, ~, k, 1), as U runs through a covering of M by coordinate neighborhoods.

2.3. An isometry / : (M , Q) -~ (M t , Q') is a diffeomorphism [: M--~ M' such that Q~(X, Y) = Q~1(~)(/.X,/.Y) for every p ~M and all X, Y ~Mr, where / . denotes the induced maps on tangentspaces. The collection of all isometries of (M, Q) onto itself is a LI~ group I (M, Q); (M, Q) is homogeneous if !(M, Q) is transitive on the points of M.

2.4. We will now consider only those pseudo-RI~MA~Nian manifolds where the metric has the same signature at every point. I f each Q~ has signature

h 2 ,~ 2 -- Z x~ -~ x~, then (M, Q) will be denoted M~.

1 h + l

2.5. A pseudo-RIElV~AN~ian covering is a covering g : N~--~ M~ of connected pseudo-RIEMA~Nian manifolds where ~ is a local isometry. A deck trans- formation of the covering (homeomorphism d : N~ -~ N~ such that ~. d = ~) is an isometry of N~. Our main tool is :

Theorem. Let ~ : N~ --~ M'~ be a pseudo-RIElVL~ian covering, let D be the group o] deck trans/ormations o/the covering, and let G be the centralizer o / D in I(N~). 1/ M'~ is homogeneous, then G is transitive on N~. I / G is transitive on N'~ and the covering is normal, then M~ is homogeneous.

The proof is identical to the proof of the R~EMA~ian case [6, Th. 1].

2.6. Our definitions agree with the usual ones in the R ~ a ~ - ~ i a n case, where each Q~ is positive definite.

3. The model spaces R~

Given integers 0 ~ h ~ n, we have a pseudo-RIE~_AN~ian metric Q(h) on a

the n-dimensional real number space R ~, given by ~O (h)~/X, Y) = - 2: x~y~ -4- n 1

+ Xxr162 where X , Y e(Rn)r correspond respectively to ( x l , . . . , xn), h + l

(Yl . . . . , yn)~ R ~ under the identification of (R~)r with R ~ by parallel translation. (R% QCa)) is denoted R~.

Let 0 a (n) denote the orthogonal group of the bilinear from Q(0 h) on Rn; let N Oh(n) be the corresponding inhomogeneous group, semidirect product

4) Th i s was b r o u g h t to m y a t tengion b y S. HELOASO~r.

8 0~] t vol. 86

116 JOSEPH A. WOLF

Oh(n).R n, consisting of all pairs (A, a) with A ~ Oh(n) and a E R ~, and acting on R n by (A ,a ) : x - , A x + a. I t i s e a s y t o s e e ](R2) ---- NO~(n).

The geodesics of R~ are just the straight lines. R~ has constant curvature zero, by a result of K. NoMIzu [4, formula 9.6], because it is a rcductive homogeneous coset space of an abelian group R n, with an R ~ -- invariant affine connection.

4. The model spaces $2 and H~. h n+ l

4.1. Given integers 0 < h < n, let Q2 denote the quadric -- 27 x~. + 2: x~ = 1 4-1-1

---- 1 in R n+l. Q2 is homeomorphic to R h X 5n-h (Sin __ m-sphere). The ortho- h n + l

gonal group of the bilinear form -- 27 x~yj + 2: xjy~, denoted Oh(n + 1), 1 h+ l

is transitive on O~. The isotropy subgroup at po = (0 . . . . . 0,1) is Oh(n). We will regard Q2 as a symmetric coset space O h (n + 1)/0 h (n), the symmetry at po being given by (x 1 . . . . . xn, xn+l) -+ (-- xl . . . . . -- x~, X~+l).

4.2. Lemma. Let f3h(n + 1) and 53h(n) be the LI~ algebras o/ Oh(n + 1) and Oh(n), and let Eij be the (n + 1) • (n + 1) matrix whose only nonzero entry is 1 in the (i, ~)-place. Then ~)h(n-+ 1) has basis consisting o] all Xi j = Et~ -- Ej~ /or h < i < ~ ~ n + 1, all Y ~ = E ~ -- E ~ for 1 < u < < v < h , andat l Z u ~ - - - - E ~ + E ~ /or 1 < u < h < ] < n + 1. ~ a ( n + 1)

B ( X , Y ) ---- - - 2 ( n - - 1 ) { - - 27 x~y~ -- 27 x ~ y ~ + 27 x~ jy~ }

where X = Z x~jX~j + 2: x , ,~Y~ + Z x,,~Z~j and Y = 27 y~jX,j + 27 y~Y~,~+ + Xy , ,~Z~ . ~3~(n) has basis {X~j, Y~,~, Z~,j}j<,, and is the eigenspace o/ + 1 /or the symmetry at To ; the eigenspace ?~ o/ -- 1 /or that symmetry has basis consisting el all X~ = Z , , ~+1 /or 1 < u < h and all X~ = Xj , ~+1 /or h < ] < n .

Proo/. The symmetry at po acts on i3h(n + 1) as conjugation by ( - - I " )

1 ; now the statements on eigenspaces and bases are clear. The proof

of the statement on the KmLI~G form is essentially due to S. I:IELOASOg [2, I,emma 6, p. 256], but is given here for the convenience of the reader. Let 15 be the eomplexification of ~h (n + 1); we have an isomorphism / of 15 onto the LIE algebra ~ (n + 1, C) of the complex orthogonal group given by

X~ -~ X~, Y~-+ Y,o and Z ~ -+ V- 1 (E~ - E~) . f3 (n + 1, C) has Krr.l~tl~o form B ' ( X , Y) = (n -- 1) trace ( X Y ) , and trace (XY) = trace (/(X)/(Y)), whence 15 has Krrx,r~lo form Be(X , Y) = (n -- 1) trace ( X Y ) . As B is the restriction of B e, our assertion follows. Q . E . D .

Homogeneous Manifolds of Cons tan t Curva tu re 117

4.3. Ident i fy ~ with the tangentspace to Q~ at pc; this gives an h n

ad (O h (n))-invariant bilinear form K-~Q~o (X, Y) ~ K-~( - X x~ y~, -4- ,V, x~y~) 1 h + l

on tha t tangentspace, where 0 ~ K ~ R, and X ~ Z x ~ X a and Y ~ XybXa 1 1

are elements of ~ . The invariance follows from the fact (Lemma 4.2) tha t K-1Q~~ is proportional to the restriction of the KrLL~SG form of ~ a ( n -k 1). Thus K-~Q~ defines an Oh(n ~-1)-invariant p s e u d o - R ~ , ~ i a n metric K-~Q on Q~.

Lemma. The pseudo-RIEMA~IAN mani/old (Q~, K-~Q) has constant curvature K =/: O.

Proo/. The manifold is homogeneous, so we need only prove our s ta tement at To. According to K. Noy~zu [4, Th. 15.6], the L~w-CIvITA connection is the canonical affine connection on Q~ induced by its structure as a symmetr ic coset space, whence [4, Th. 10.3 and 15.1] the curvature tensor is given by U-~(X, Y).Z ---- -- [[X, Y], Z] for X , Y, Z E ~ . We choose local coordinates

0 ---- Xb, 1 ~ b ~ n. In these coordinates, (u b} at pc such tha t ~ ~o _ _

qb~(Po) -~ K-~Q~o (XD,X~) ~ -- K-XOb~ if b ~ h, K-l~b~ if b > h, where b~ is the KRO~CKER symbol. Also, RbCcb--~ 1 ~ --R~r for c # b > h ,

Rb~b~ = 1 ~ -- RbC~b for c ~ b _~ h, and all other Rdeb~ are zero, a t p o. Thus R ~ b 0 = K - x = - R b ~ b for c g h < b and for b ~ h < c , and R~b~----

- - K - l = - R b ~ b if b :~c and either b , c > h or b ,c ~ h , and all other R~b~ are zero. This gives Rd~b~(po) = K (q~b(Po)q~(Po) -- qdb(Po)q,,(Po)}, whence (Q~, K-~Q) has constant curvature K . Q . E . D .

4 .4 . We define S~h : (Q~,Q) and H ~ ~- , . , -h (Q~ - Q ) Note t ha t the signatures of metric are correct. S~a has constant curvature -k 1 ; i t is the indefinite metric analogue of the sphere S n ---- S~o. H~ has constant curvature -- 1 ; it is the indefinite metric analogue of the hyperbolic space H n ---- H~. We will sometimes speak loosely and refer to (Q~, K-1Q) as 82 if K > 0 or H~_ a if K < 0 .

and / ~ - a are the connected, simply connected pseudo-RIE~AN~rian

manifolds corresponding to S~a and H'._h �9 S-~ ---- S~a and /-7~_ a : H n._` i f

h < n - 1; 5"~n-1 and / ~ are the respective universal pseudo-RI~iANNian

covering manifolds of S~_l and H~ ; ~ and H~o " are the respective components of Pc in 5~n and H i .

118 Josze~ A. Wo~F

4.5. Lemma. I(S~h) = Oh(n -f- 1) = H ~ I( .-h).

Proo/. The isotropy subgroup at :re is maximal for conservation of the metric. Q . E . D .

4.6. We mention an alternative description of SN and HN. The alternative description is more direct, but the description in w 4.1 tc w 4.3 simplifies the proofs of Lemmas 4.3 and 4.5.

One can prove tha t the manifold SN of curvature K :> 0 is isometric to h n-I-1

the metric submanifold of RN +l given by -- 2: x~ + 27 x 2 -~ K-�89 and that the 1 h + l

manifold H~ of curvature -- K < 0 is isometric to the metric submanifold of h-I-1 n-I-1

R~+~ given by -- X x~ ~- 2: z~ = -- (g-~) . 1 h+2

5. The universal covering theorem

Theorem. Let M~ be a complete connected 1oseudo-Rx~L~NNian mani/old el constant curvature K. Then, assuming the metric normalized such that K is -- 1 ,0 or -t- 1, the universal pseudo-R~MA~ian covering manifold o/ M~

is H~ i/ K < O, R~ i/ K : O , and S~ if K > O.

Proof. The universal pseudo-RIEMANNian covering manifold N~, of M~ is complete, connected and simply connected; we must show N~ is isometric to

H~,/~A or ~'~. Let A~ be the one of H~, R~, S~ of constant curvature K. Let a eA'~,b e N N and let f , : (A'~)~---~-(NN) ~ be a linear isometry of

tangentspaces; 1, exists because the metrics have the same signature. Given a broken geodesic a of A N emanating from a, let fl be the broken geodesic on N~ emanating from b which corresponds (see[3]) to a under ] , . Let S be a 2-dimensional subspace of (A~)~ ; the curvature tensors, hence the curvature forms, take the same values on corresponding parallel translates of S and ] ,S along a and fl, as they are determined by the sectional curvatures. The torsion forms are identically zero because we are dealing with LEW-CIVITA connections. I t follows from a theorem of N. HICKS [3, Th. 1] tha t ] , extends to a connection-preserving diffeomorphism g : A N -+ N N. Let a' ~ A~, let s be an are in A~ from a ~ to a, and let s , be the parallel translation along

from a' to a; let t , be the parallel translation along g(s) from b to b'-=g(a'). The tangential map (g,)~,, equals $ , . f , . s , because g is connection-preserving; as t , .1, .s, is a linear isometry of tangentspaees; it follows t h a t g is an isometry of p s e u d o - R I ~ - i a n manifolds. Q.E.D.

Homogeneous Manifolds of Constant Curvature

Chapter II. Homogeneous manifolds of nonzero curvature

119

The goal of this Chapter is the global classification of the homogeneous p s e u d o - R i f f i a n manifolds of constant nonzero curvature. The main tools are Scm~R's Lemma, real division algebras, bilinear invariants of group representations, and the technique of changing the basefield of a quadrio.

To a manifold M~ of constant curvature k, there corresponds a manifold hr,"_h of constant curvature - - k , obtained from M~ on replaeing the metric by its negative. For complete manifolds, this correspondence is given by

~'~/D~---~ H'~_h/D. I t suffices, then, to do our classification for the case of positive curvature.

6. The real division algebras

We assemble some facts on real division algebras, for the most part well known, which will be useful in the sequel. All algebras will be finite dimensional.

6.1. A real division algebra f is one of the fields R (real), C (complex) or K (quaternion). Let x-> x denote the conjugation of F over R, let f* denote the multiplicative group of nonzero elements of F, and let F' denote the subgroup of unimodular (xx ----- 1) elements of F. dim. F is understood to be the dimension of F over R. A standard R-basis of F is an R-basis {al,. . . ,adim. F} such that a l = l = - - a ~ for i > l , a ~ a j - ~ - - a ~ a , for i : # i , i > 1 , ~ > 1 and 4 -a i a ~ e{ak} for each i ,~ . Given x e F , Ix ] i s the positive square root of x x.

6.2. /=' is a compact topological group, so any discrete subgroup is finite. The finite subgroups of R ~ are 1 and {4- 1 }. The finite subgroups of C' are the cyclic groups Z~ of any finite order q > 0; Za is generated by

exp (2 ~V "-~- 1/q). The finite subgroups of K' are cyclic or binary polyhedral. We recall the binary polyhedral groups. The polyhedral groups are the

dihedral groups ~,~, the tetrahedral group ~ , the octahedral group s and the icosahedral group ~ - the respective groups of symmetries of the regular m-gon, the regular tetrahedron, the regular octahedron and the regular icosahe- dron. Each polyhedral group has a natural imbedding in 8 0 ( 3 ) . Let 7t : K ~ -+ 30 (3 ) be the universal covering; the binary ~olyhedral groups are the binary dihedral groups 3 " = ~-1 (~,~), the b/nary tetrahedral group ~s = = ~t-l(~), the binary octahedral group ~ * = zt-x(i3), and the binary icosahedral group ~* = ~r -1(~). We exclude ~ because it is cyclic. Note that isomorphic finite subgroups of K' are conjugate.

120 Jos~.P~ A. W o ~

6.3. Lemma. Let D be a subgroup of F* and set D' ~ D ,~ f ' . Then D is a discrete subgroup of f* if and only i/ D' is finite and D ----- {D', d} /or some d ED with [d[ ~ 1 .

Proof. Suppose D discrete. I f D ----- D ' , set d = 1. Otherwise, note tha t D i ----- {[ g ] : g cD} is a discrete subgroup of R*, for D cannot have an infinite number of elements in a compact neighborhood of F . I t easily follows tha t /)1 is generated by an element [ d [ which minimizes [ d [ -- 1 > 0. To see D -~ {D', d}, we note tha t every g e D satisfies [ g [ ---- [ d[m for some integer m; thus g d - ~ e D ', so g ~ {D' ,d} .

The converse is trivial. Q.E.D.

7. The 6 classical families of space forms

7.1. Let O ~ s < m be integers, choose a basis {zj } in a left m-dimensional vectorspace V over f , and consider the hermit ian form QF(u, v ) =

$ m

= - - X u ~ v ~ X u ~ v ~ where u = Z u ~ z ~ and v = Xvjz~. The real part 1 s + l sr m r

Q(u, v) = ~ ( Q p ( u , v)) is given by Q(u, v) = - Xx~yj § Zxcyr where 1 s r + l

r = dim. F, (a~} is a s tandard R-basis of F, {w~) is the corresponding R-basis of V, given by aiz j -~ w~(~_l)+~, u = Xxjw~ and v ~ Xy~w~. As QF(u, u) is real, this allows us to ident ify Q~r-1 with the hermit ian quadrie of equat ion Qe(u, u ) = 1. Under this identification, both F' and the un i ta ry group U'(m, F) of QF becomes subgroups of the orthogonal group O~'(mr) of Q. The actions of F' and U*(m, F) commute because

O m'-i and F' acts freely US(m, F) is F-linear, U'(m, F) is transitive on - s t , on Q~,-1. I f D is a finite subgroup of F ' , i t follows tha t S~/-1/D is a homogeneous p s e u d o - R I ~ . ~ z r manifold Me~ r-1 of constant curvature q- 1. We will refer to these manifolds as the classical space forms; special cases are studied in [5] and [6].

7.9. Le t o f denote the family of connected classical space forms for F = R. More precisely, to every triple (h, n , t) of integers with 0 g h ~ n , 1 ~ t < 2, and t ----- 2 i f h = n , there corresponds precisely one element S~a/Zt of ~ . Note t ha t S'~/Z~ is isometric to a component of S~. The elements of o f are the spherical and eliptic spaces of indefinite metric.

Le t ~ " denote the family of classical space forms for F ---- C which are not in ~ . To every triple ( h , n , t ) of integers wi th 0 ~ h _ ~ n , t > 2 , h even and n - b 1 even, there corresponds precisely one element S~h/Zt of ~ ' . The elements of ~ " are the cyclic spaces of indefinite metric; they correspond to the IAnsenr~ume of THRELrAT,!. and SEIFERT [5].

H o m o g e n e o u s Manifolds of C ons t an t Curva tu re 121

Let ~ denote the family of classical space forms for ]: = K and D binary dihedral. To every triple (h, n, t) of integers with 0 ~ h ~ n, t > 1, and both h and n + 1 divisible by 4, there corresponds exactly one element Sh/~) t of ~ . The elements of ~ are the dihedral spaces of indefinite metric; they correspond to the Prismar~ume of Tm~ELFATX, and SEI~I~T [5].

Let ~ , (~) and d~ denote the families of classical space forms for ]: -- K, with D respectively binary tetrahedral, binary octahedral and binary icosahedral. To every pair (h, n) of integers with 0 ~ h < n and both h and n + 1 divisible by 4, there corresponds precisely one element 8~/7s of c~, one element S'~/~)* of (~, and one element 8~/3" of (~'. These are the tetrahedral octahedral and icosahedral spaces of indefinite metric; they correspond to the Tetraederraum, Oktaederraum and DodeIcaederraum of THRELFALL and SmFEI~T [5].

8. Behavior of quadrie under field extension

We will consider material converse to that of w 7.1 and discuss reducible groups which are transitive on quadrics.

h n + l 8.1. Lemma. Let Q be the bilinear form Q(u, v ) - ~ - I u ~ v j + l u ~ v ~

1 h + l on V = R n+l, so Q~ isgivenby Q(u, u) = 1, and Oh(n + 1) istheorthogonal group of Q. Let G c O h(n + l) be transitive on Q'~, let ]: be a real division subalgebra of (~ ( V) which centralizes G, and let r ~- dim. ]:. Then ]:' c O h ( n + 1),s----h/r and m = (n--f- 1)/r are integers, and there is a Q-orthonormal R-basis {wj} of V and a standard R-basis {aj} of ]: such that wrj+~ -~ a~w~j+l, {wl, w,+l,. . . , Wr(m-1)+l} iS an ]:-basis of V, and Q(u, v) -~

$

= c~e (-- Z x j y j -l- l x j y ~ ) where u ---- l x j w ~ , _ l ) + l and v -~ 2:yjw, cj_l)+l 1 8+1

with x~, y~ ~ F.

Proof. Let a e ] : ' . Given u , v E Q~ we have some g ~G with g ( u ) = v , so Q(av, av) = Q ( a g u , a g u ) - - Q ( g a u , g a u ) = Q ( a u , a u ) ; thus f ( a ) = Q(av, av) for any v ~Q~ is well defined. Let w ~ V with Q ( w , w ) > O . There is a real x > 0 with x w EQ~, so Q(aw, a w ) ~ x - 2 Q ( a x w , a x w ) = x-2/(a) = f(a)Q(w, w). Note tha t /(a) > 0: this is clear for ]: = R; ]:' is connected if ]: :/= R, so we need only check that f(a) =/= 0. As /(a) = 0 would imply that a is a homeomorphism of the open (n + 1)-dimensional set P = { w ~ V : Q ( w , w ) > 0 } into the lower dimensional set L = -~ {w E V : Q ( w , w ) = 0}, this is clear. Now let a , b ~]:', and note that (v ,Q~)/ (ab) = Q ( a b v , a b v ) = f(a)Q(bv, b y ) = / ( a ) f ( b ) ; thus / is a homo- morphism of the compact group ]:' into the mnltiplicative group of positive real numbers; it follows that f (F) = 1.

122 J o s ~ , A. WOLV

We now have Q(w, w) ~ Q(aw, aw) for a ~ f ' and w ~ P . B y cont inui ty of Q(aw, aw) - Q(w, w), this also holds if w is in the boundary L of P . Polar izat ion shows Q(au , a v ) = Q ( u , v ) i f u , v and u ~ v lie in P ~ L . Now let {vj} be a Q-orthonormal R-basis of V, set uj-----v~ for ~'> h,

and set uj - - vj ~ V2vn+ 1 for i <: h. One easily checks tha t {uj} is a basis of g such t ha t u~, u~ and u s ~- uj ~ P ~ L for every (i, j) ; i t follows tha t Q(au, av) : Q(u, v) for a E F' and u , v e V. Thus F' c Oh(n ~ 1).

Le t {a~} be any s tandard R-basis of /:, and choose w I ~ g such tha t Q ( w l , w l ) - - - 1 i f h : > 0 , Q ( w l , w l ) : 1 if h - : 0 , set w~----a~w 1, and let F1 be the Q-orthogonal complement of { w l , . . . , wr }. Choose W,+l ~ V1 such tha t Q(w~+l, w~+0 -- - 1 i f h > r , Q (w~+l, W~+l) : - 1 if h : - r , set w~+~ =- -~ a~wr+l, and let Vs be the Q-orthogonal complement of { w l , . . . , w~}. Continuing this process, we obtain the desired basis and see tha t s and m are integers. Q . E . D .

8 .2 . In the following, we will not always ment ion Q explicitly. However , a to ta l ly isotropic subspace will mean a to ta l ly Q-isotropie subspace (one on which Q vanishes identically), o r thonormal will mean Q-orthonormal, and or thogonal <<complements>> will be t aken re la t ive to Q unless we s ta te otherwise.

Lemma. Let G c 0 h(n ~- 1) be transitive on Q'~, let U be a proper G- invariant subspace o/ V ---- R n+l, set W : U ~ U • and let k : n ~ 1 - - h. Then 2h ~ n , W is a G-invariant maximal totally isotropic subspace o/ V , G acts irreducibly on W, and V has an orthonormal basis {w~} such that (e~ -~ wa+ j + w~ /or 1 ~ ] g k) {e~ . . . . ,e~} is a basis o/ W and {w~+ 1 . . . . , wa, e l , . . . , e k} is a basis o/ W x . Finally, i/ F is a real division 8ubalgebra o/ ~ ( V ) which centralizes G and preserve~ U, then W and W • are/:-invariant and we may take {wj } to 8atis/y the conclusion o/ Lemma 8.1.

Proo/. I f /: is no t given, set F----R. As U and U • are proper G-invari- an t subspaces of V , and as Q~ contains a basis of V, i t follows t h a t Q is negat ive semi-definite on U and on U • . Were Q nondegenerate on U, we would have V ---- U G U • with Q negative definite on each summand, so Q would be negat ive definite on V. Thus Q is degenerate on U, so W r 0. Now W is a proper G-invariant to ta l ly isotropic subspace of V, F-invariant because F c Oa(n ~- 1) (by L e m m a 8.1) implies t h a t U and U • are F-invariant . Thus we m a y assume W---- U. W and V are left vectorspaces over F. Set Q(v) = Q (v, v) for v c V.

L e m m a 8.1 gives an F-invariant or thogonal decomposit ion V - : V 1 (~ Vs with Q negative definite on V~ and positive definite on Vs. Choosing an /:-basis { b l , . . . , b , } of W, we have b~ = r~ @ s~ with r~ ~ V~, s~ ~ Vs.

Homogeneous Manifolds of Constant Curvature 123

Set e~=pl -~q l -~Q(s~) - � 89 with P~ ~ V~,q~ ~ V~. We construct e~---- = p~ ~- q , with P~ ~ V1, qu ~ V2 from (e~ . . . . . e,_l, b~} b y setting z ~ - ~ b , , - - Z Q ( q ~ , s ~ ) e ~ - = x ~ - ~ y , , with x ~ V ~ , y ~ V ~ , and by %----

i<u = Q(y,)-�89 Now ( P i , - ' ' , P~; q l , - ' ' , q~} is l inearly independent over F, Q(p~, q~) --- o, and Q(q~, q~) -~ ( ~ = - Q(p~, p~). By Lem m a 8.1, we can complete it to an or thonormal ]:-basis (p~ . . . . . p~, c 1 . . . . , c ~ ; q~ . . . . . q~, d ~ , . . . , d~ of V. W ~ then has ]:-basis (e~ . . . . ,e~,c~ . . . . . c ~ , d ~ , . . . , d b } and Q is negat ive semi-definite on W • , so b = 0. Now let r----dim, fi and let {a~} be a s tandard R-basis of F. Set w~(j_~)+l = pj for for 1 ~_~ ] ~ a , wr( t . . l_aTj_l )_ t_ 1 = ql for 1 g j ~ t, L e m m a is now easily verified.

1 <: j ~__ t, Wr($+j_l)+l = (~j and w~+t ---- a iwr t+ l . The

Q . E . D .

8.3 . Lemma. Let G c 0 a (n ~ 1) be transitive on Q'~, let U and W be proper G-invariant subspaces o/ V = R ~+1, and suppose U ~ W = O. Then 2h ----- n --~ I, U and W are maximal totally isotropie subspaces o/ V, and V = U |

Proo/. Let k - - - - n - ~ 1 - - h , U r - - U ~ U ~ and W ~ - W ~ W I . By L e m m a 8.2, bo th U' and W f are maximal to ta l ly isotropic and are k-dimensional, so we need only show V = U' O W t. Were Q(U ~, W ~)-= o, U ~ O W ~ would be to ta l ly isotropie, contradict ing the fact t h a t U ~ is maximal to ta l ly isotropie. I t follows tha t U / �9 W' has an element v with Q (v, v) = 1, so Q~ c U' �9 Wt because G is t ransi t ive on Q~ and preserves U ~ (~ W I. Thus U ' O W ' = V. Q . E . D .

We will now consider the existence o f a G c Oa(n ~ 1) which is t ransi t ive on Q~ and preserves two l inearly disjoint proper subspaces of V.

8 .4 . Given integers 0 "g s ~ m, we consider the hermit ian form Qp(u, v) =

- 2 7 u j v j ~-2:u~vj on a left m-dimensional vectorspace V over F. As 1 s+l

before, r - = d i m . F , Q ( u , v ) - - ~ d ~ e (QF(u,v) ) , and Q ~ - I is given b y

Qe(u, u) = 1. Let U be an F-subspace of V. I f U is to ta l ly Qe-isotropie, it clearly is

to ta l ly Q-isotropic. Suppose, conversely, t ha t U is to ta l ly Q-isotropie, let u , v c U, and choose a s tandard R-basis (a~) of F. Qy(u, v) -= X~alx ~ with x ~ e R ; o - - - Q ( a ~ u , v ) - ~ - x i shows Q e ( u , v ) - ~ o ; thus U is to ta l ly Q~-isotropic.

The un i t a ry group U~ F) of QF is the centralizer of F in OS'(mr). One inclusion is obvious and has a l ready been noted, so we need only show t ha t an element g e Or(mr) which centralizes F' lies in U'(m, F). Central- izing F , g is F-linear. Le t u , v ~ F and note d~e (QF(a~u, v)) = Q(a~u, v) ~-

124 Jos~P~ A. WOL~

: Q(ga~u, gv) = Q(aigu, gv) = ~ e ( Q F ( a ~ g u , gv)) . I t follows t h a t

Q~,(u, v) : QF(gu, gv), so g E U'(m, F). Combin ing these two fac ts wi th L e m m a 8 .3 , we see t he usefulness of :

$ 2 5

L e m m a . Consider the hermitian /orm QF(u, v) ~ -- l ujvj -~ l u/v~ on a 1 s + l

le/t 2s-dimensional vectorspace V over F, let U and W be totally Qp-isotropic F-subspaces o] V such that V ~ U 0 W. Then there are Fbases {ei} o/ U and (/j} o[ W with QF(e~, /~)~ 23~. Let ] represent the general linear

(o o) group GL(s , F) on V b y ' [ ( ~ ) = ~-1 in the basis {e~,/~} o/ V, and

let G = {g �9 US(2s, g) : Ug : U, Wg = W} . Then G is the image o/ f , and G is transitive on QF(u, u) = 1 i / and only i/ F = R.

Proo/. The bases {e,} a nd {[j} are eas i ly c o n s t r u c t e d b y t he m e t h o d o f L e m m a 8 .2 . I t is easy to check t h a t G is the image o f [ , a n d the las t s ta te- m e n t is obv ious for s = 1. N o w as sume s > 1, a~d we'l l p rove the las t asser t ion. F o r 1 G i G s , set w j = � 8 9 and w~+~= � 8 9 so {wa} is a Q F - o r t h o n o r m a l basis o f V over F. As QF(w,+I, w~+l) --~ 1, we see t h a t G is t r ans i t ive i f a n d o n l y if, g iven x �9 V wi th QF(x, x) = 1, t he re is an a �9 GL(s, F) wi th / ( ~ ) : w ,+ t -~ x . L e t x ---- Xx~w~; i t is easy to check

t h a t [ (~) : w,+l --> x i f a n d o n l y if

As ~ - i = I , th is wou ld give 1 = X(x~+~ § x~) (x,+~ -- x~) = -- X x~7~ + 1 1

$ - - 8 $ 8

1 # 1 # ~ 1 1

then gives Zx~x.+~ ---- ,Vx,+~.~, so 17x~x,+~ �9 c ~:. If F ~ R we can 1 1 1

a lways f ind an x ~ V wi th Q F ( x , x ) - - - 1 a n d x~x~+~r so G will n o t

be t r ans i t ive on Q~(u, u) = 1. W e n o w as sume F ---- R, wr i te Q -~ QF, iden t i fy ~O ~'-~ wi th QF(u, u) = 1,

a n d will show G t r ans i t i ve on ~,~)~ -~ . W e se t z -~ x~+ 1 ~- z i , y = x~+ i - - x~, r -~ ---- (x,+~ q- x ~ , . . . , x~, q- x,) , a n d t ~ (x,+~ - - x ~ , . . . , x ~ , - - x,) . W e wish

to find ~ = tu E G s wi th r162 . I f y : / = 0 , we se t

r162 _ y - l . t t a n d h a v e ~-1___ I - - ~ t . r . I f z : ~ 0 , we se~

~ - ~ z - l ' r a n d h a v e ~ = - - z t . t I - - ~ t . r " N o w suppose

x - - - - - z = 0 , n o t e t h a t a so lu t ion wou ld give ~ - t = A Lt t u . v + A . B '


pu t the usual euclidean inner p roduc t (a, b) on the space R ' - I o f rent (s --1)- tuples, and observe tha t r , t, u, v e R '-1. Le t X be an (s -- 3)-dimensional subspaee of r • ~ t • _1_ taken relat ive to ( , ), and let {Y3 . . . . . Y~-I} be an or thonormal base of X . (r, t) ~- Q(x , x) -~ 1, so we can ex tend {yj} to bases {e,y3 . . . . ,Y8+1} of t • and (d, y3 . . . . . Y~-I} of r • with ( d , e ) : l and (d, yj)---- 0---- (e,y~). Now pick ui, v i E R with ulv l ~ u ~ u ~ 1. We set B---- (ul~d, u~d,ty3, . . . , rye_l ) and ~A ~ (vlte, v~te, ty3 , . . . , rye_l), so r . B ~ - 0 and A . t t ~ - 0 by construction. Set u----- (u2, -- u~, 0 , . . . , 0) e R s-1 and v : ( v 2 , - - v l , 0 . . . . ,0) eR '-1; l : u l v 1Jru2v ~ gives t u . v ~ A . B - ~ I , so the desired ~ and ~-1 in GL(s, R) are constructed. Q . E . D .

8 .5 . We end w 8 with two more lemmas, placed here for lack of a be t te r location.

Lemma. Let G c O h (2h) be transitive on Q ~ - I , let U and W be proper G-invariant subspaces of V ~ R 2a with U ~ W = O, and let Gv be the restriction o/ G to U. Then there is no nonzero Gv-invariant bilinear /orm on U.

Proo]. Let B (u , v) be a Gv-invariant bilinear form. I f B is not skew, we m ay replace it with B ' ( u , v ) : B ( u , v ) ~ B (v , u), which is symmetric . But the proof of L e m m a 8.4 and the i rreducibi l i ty of Gv shows t h a t Gu has no nonzero symmetr ic bilinear invariant ; we conclude tha t B is skew. Assume B =/= 0, so B is nondegenerate by the irreducibil i ty of Gv. Lem m a 8.4 gives us an or thonormal basis (v i} of V such tha t (e i -~ vh+ ~ + v i and ] i : v h + ~ - - v ~ for 1 ~ i ~ h ) (e~} i s a b a s i s o f U and {/~} a b a s i s o f W,

and every g ~G is of the form g = ~g_~ in the basis {e~,[~} of V.

~gt_~ : gl preserves B which, containing

G, is t ransi t ive on Q~h-1. 11

to be of the form gl ---- / 0

10

Let G" be the subgroup of G' given b y taking gl 0 . . . . 0 )

, . One checks dim. G' = �89 A- 1) and gl

dim. G" ----- �89 (h -- 1) -- 1 ; observing tha t G" is the isot ropy subgroup of G' at v~+ 1 c Q~h-1, it follows tha t h -- 1 : dim. G' -- dim. G" ----- dim. Q~h-1 ~ 2 h -- l , so h ~ 0. This contradicts B =fi 0. Q . E . D .

8 .6 . Lemma. Let D be a subgroup o[ O h (n-f- 1), let G be the centralizer o] D in Oa(n - k 1), suppose G transitive Q~, and let U be a maximal totally isotropic G-invariant subspace o/ V -~ R "~+1. Then U is D-invariant.

Proo]. Let d E D. Then dU is G-invariant and, as (7 is irreducible on U, ei ther dU----- U or dU~, U - ~ O . In the la t te r case, 2 h ~ n q - 1 and

126 Jossr~ A. WOLF

V ~ U (~ dU by Lemma 8.3; B(u, v) ~ Q(u, dr) is a Gv-invariant bilinear form on U, so B-----0 by L e m m a 8.5. Thus Q(U, dU)----O so Q ~ - o , contradicting dU ,-, U = O. Thus dU = U. Q. E. D.

9. The 8 exceptional families of space forms

9.1. In an orthonormal basis {v~ } of V = R 2h , we have

(cosh (8)Ia sinh (8)I~ ~ Oa(2h) ' rH(s) = \sinh (s)I a eosh (s)Ih]

where s > 0 is a real number and I h is the h • h ident i ty matrix. To check this, we set e j = v h + j-i-v~ and / j = v h + j - r e for 1 _~j _~h, observe tha t

the hyperbolic rotat ion rH(S)-~ (exp (--0 s)Ia exp0(s)Ih) in the basis

{/~, e~} of V, and refer to the first par t of L e m m a 8.4. L e m m a 8.4 also tells us t ha t the centralizer G of r~i(s) in Oh(2h) is transit ive on Q~a-~. Note t ha t rH (s) is conjugate to rB (t) i f and only if s ~- t, t also being taken positive, and tha t for a given s the subgroups ~ ( s , + ) ---- {r/t(s)}, .~(s, --) : : {-- r H(s)} and ~ ( s , d-) : {-- I~h, rH(s)} are discrete subgroups of Oa(2h). I t follows tha t these groups are free and properly discontinuous on Q~a-1, and tha t the quotients S2aa-~ /.~ (s, +), S~a-~ /~ (s, --) and S~a-1/~ (s, :]:) are homogeneous manifolds of constant positive curvature. The class ~ of these hyperbolic-rotation spaces is our first class of exceptional space forms. To every triple ( h , s , ~ or -- or 4-) where h > l is an integer and s : > 0 is real, there corresponds exact ly one element S2ah-1/~(s, + or -- or • of ~ , and distinct elements of ~ , are not isometric. Note tha t

( s , + ) ~ Z ~ . ~ ( s , - - ) and .~(s, • • Z2.

9.2. In an or thonormal basis {v~} of V = R n+l with 2 h > n and k ----- n ~- 1 -- h even, we choose a skew nonsingnlar k x k matr ix d and have

tB(d) = 0 Ih_ ~ 0 c Oh(n + 1). This is best seen by setting d 0 I ~ - d

ej ---- va+ ~ + v~ and /~ ---- va+ ~ -- vj for 1 _~ ?" g k and observing 'i) t ha t tB(d) = I~_~ in the basis fl = {h . . . . , / ~ ; v ~ + l , . . . , v h ; 0 ~

e , , . . . , e ~ } of V. I f a linear t ransformat ion of V commutes with tH (d), i t must preserve the

subspaee U of basis (e~}. Thus the centralizer G of tz~(d) in Oh(n ~ 1) preserves both U and U ' , and consequently is in upper tr iangular block

form g---- g~ re la t ive to ft. For a mat r ix o f t h a t form, g.tH(d)=tH(d).g

Homogeneous Manifolds o f 0onstant , Curvature 127

is expressed b y dgs = gld, and g c O h (n -f- 1) is given b y g8 = tgl-1, g4 c O (h - - k) o rd inary or thogonal group, 2<(g1)i, (ga) j ) -'~ 2<(ga)~, (gl)j> --~ <(g~)~, (g2)~) and 2<(g0i, (gs)J> = <(g~)~, (g4)j> where (g~)~ denotes the v th row of g , and < , > is the ord inary euclidean pairing of m-tuples.

We first no te tha t , i f d ' is ano ther skew nonsingular k X k mat r ix , then tH(d') is conjugate to tH(d) b y an e lement of Oh(n -4- 1) which preserves U.

(~ For the associated skew bilinear forms on R ~ are each equivalent to - - I '

so there is a nonsingular k • k m a t r i x gx wi th gldtgl = d'. Now set

g = I and notice g . t H ( d ) g - I = t H ( d ' ) , so the subgroups 0 tg1-1

3 ; ( + ) = {tH(d)}, 3 ; ( - - ) = {-- tH(d)} and 3 ; ( + ) = { - - I .§ of Og(n -t- 1) are defined up to eonjugacy wi thout reference to a n y par t icular d. These subgroups are free and proper ly discontinuous on Q~; we will see t h a t the corresponding quotients of ~ are homogeneous.

k h - - k

We m u s t show G t rans i t ive on Q~. We take a n y x = Z a~v~ -F Z, bjvk+ ~ -b k 1 1

+ Zcjvh+~ EQ'~, and wish to find g ~G wi th g(vh+l) = x. Calculating 1

g (vh+0, we see tha t this is equivalent to c - - a ---: (91)1, c + a ---- (93)1 + (ff6)~ and 2b : (g~)l, where a : (a~ . . . . . a~), b : ( b l , . . . , bh_~) and c = = (ca , . . . , %), and we add coordinatewise. As H = {91 ~ G/_(k, R): tg ldg l=d} is the real symplet ic group, t rans i t ive on the nonzero elements of R k, we m a y choose gl r H wi th c - - a = (901. g6 : ~g1-1 is then determined, and (g~)l and (ga)l are g iven b y gvh+l : x. We m u s t fill out g2 and ga, and then cons t ruc t g4 and gs. Suppose we have the first u - - 1 rows of g2 and 9a, 2 < u ~ k, and 2<(gx)i, (g3)r + 2 <(g3),, (gx)j> = <(g2),, (g2)> for i , j < u , with (g,), = 0 for l < i < u . Set (g2), = 0; the conditions on (ga)~ are then <(gl)~, (g3)~> § <(g3)~, (gl)~> = 0 for 1 ~ i < u, so we have hnear forms p~ on R * and constants q, such t h a t these conditions are p~ ((ga)u)A- § q~ = 0 for 1 < i < u. The rows of g~, hence the p~, are independent ; thus w e h a v e a s o l u t i o n (g~)~. Thus 9, and g3 are constructed. Set ga = I~_ , , and the existence of ga follows, as above, f rom the independence of the rows of gl. Thus G is t rans i t ive on Q~, so ~ /~ . ( - f - ) , S '~ /3 ; ( - ) and S~/3;(~:) are homogeneous manifolds of cons tan t posi t ive curvature . The class ~ t of these hyperbolic-translation spaces is our second class of except ional space forms. To every tr iple (h, n , § or - - or -t-), where h and n are integers wi th 2 h > n > 1 and k : rt ~ - 1 - h even, there corresponds exac t ly one e lement 5"~[3;(A- or - - or + ) of ~ . Notice t h a t 3 ; ( + ) ~ Z ~ 3 ; ( - - ) and

3;(-4-) = z • z~.

128 J o s ~ A. Wor~

9.3 . Assume 2 h > n > l and t ha t k - ~ n + 1 - - h is divisible b y 4; for eve ry complex number u ~ R, we have groups ~ (u, -~ or -- or • which give homogeneous space forms. I f ~ : ( + ) is viewed as the addit ive group of integers, t hen the free abel]an par t of ~ is the addi t ive latt ice in C generated b y 1 and u .

( 0 ~) and define J = We retain the no ta t ion of w set q : - - 1

".. e G / . (k ,R) . Now let d be any skew nonsingular element of q

G/ . (k ,R) such t ha t d J + J d = O . As j 2 = _ i ~ , we m ay ident i fy C with the subalgebra of C ( U ) consisting of all a I + b J with a , b ~R. Given u = a I + b J ~ C with b ~ 0, we define subgroups

~(u , + ) - - {t~(d), tH(dU)} ~ (u , - ) = {tH(d), -- tx(du)} ~ ( u , -4-) : {-- I~+1, tn(d) , tH(du) }

of Oh(n + 1). As dv is skew and nonsingular for v e C* and tu(dVl).tn(dv2)-~ ---- t~ (d(v I q- v2)) for v~ e C, these lattice groups are free and proper ly discontinuous on Q~. Le t r the collection of space forms 5~a/~ (u, q- or - - or • we must check tha t the elements of eL 9 are homogeneous.

Le t g = g4 ~ Oh(n + 1) in the basis fl o f V; g centralizes 0

~ ( u , + or - - or 4-) i f and only i f g6J ~ Jge and dge -~ g~d. We now m a y proceed as in the p roof t ha t the elements of ~ are homogeneous, except t h a t H mus t be replaced by H' ~ {gl c Gs R): tg~dg~ -= d, g~J ~-- Jg~} ; we need on ly check tha t H ' is t ransi t ive on the nonzero elements of R k. L e t 2s = k; we give R u the complex s t ruc ture defined b y J ; H r becomes

{gl �9 Gl.(s, C): tgl'do.gl ----- de} where d~ is the skew C-linear map x--~d(x) and conjugat ion is wi th respect to a C-basis of the form {ea~._l}, whence the t rans i t iv i ty of H ' on R k -- {0} --~ C s -- (0} is clear. Thus the lattice space /ornm 5~a/~ (u, -b or - - or 4-) are homogeneous. One can check t h a t ~ ( u , q- or - - or 4-) is defined up to conjugaey in Oh(n q- 1) wi thout reference to d or J , and t ha t (x, y = q- or -- or =t=) ~ ( u , x) is conjugate to ~ ( v , y ) if and only i f x = y and the lattices {1 ,u} and {1,v) in C are equal.

9 .4 . Assume 2h ~> n > 1, h is even and k ~ n q- 1 - - It is divisible by 4; we then have an infinite dihedral group ~)*, extension of ~ ( • by an au tomorph i sm of order 2, and, for m ----- 3, 4, 6 or 8, some extensions ~ (u) o f the lat t ice groups.

H o m o g e n e o u s Mani fo lds of C o n s t a n t C u r v a t u r e 129

( ~ '0) We retain the notat ion of w and, as before, set q=-- -- 1 ( ' ) J = ".. e GL(k ,R) , and choose a skew nonsingular k • k matr ix q ( ') d with J d + d J = O . Let J ' = ".. E Gl.(h -- k, R) and let

q Jn+l = j r relative to fl; J~+l �9 Oh( n + 1). l~inally, u = a I k +

0 -+- b J , b :~: O, represents a non-real complex number. ~)* = (J~+i, t~(d)) is an infinite dihedral group (x, y : x 4 = 1, x y x -I = y- l) .

R~ is the rotat ion cos (2~r/m)Ih+ 1 + sin (2:r/m)J~+ 1 of V. R~r Oh(n + 1). Note t h a t R4 = Jn+~. Let m = 3, 4, 6 or 8, and suppose t h a t the lattice {1 ,u ) in C is invariant under left multiplication by cos(4:r/m)

+ ~ l s in (4 : r /m) , where ]/~-- 1 is chosen such tha t u - - - -a~- ] / ~ lb . We then set ~ m ( u ) = {R,,, tH(d), tH(ud)). ~ ( u ) is free on Q~, and acts properly discontinuously because, as

R, , . t~(vd) .Rm -~ = t~ ((cos (4:r/m) + P / ~ 1 sin (4:r/m))vd)

for v E C , ~ ( u , + ) has index m or m/2 in ~ ( u ) . Similarly, ~)*~ is free on Q~, and acts properly discontinuously because ~(~-) is a subgroup of index 4. Let ~)~ be the collection of space forms S ~ / ~ (u) (m = 3, 4, 6, 8) and let ~ be the collection of space forms " * �9 Sa/~oo, we must show tha t the elements of ~f)~ and of ~ are homogeneous.

Let D = ~ ( u ) or ~*~; in the basis fl, the centralizer of D in Oh(n+ 1)

is the collection of all g = g4 g6 �9 Oh(n + 1) with gJ~+~ = J~+lg \ 0 0 ge

and gld ---- dg6. We give V the complex structure defined by J~+l, so Q is the real part of a hermit ian form Q, and (v~_ I) is a Q~-orthonormal basis of V. We set

(]~ : ~ 8 , h : 2 t ) fl�9 ~--- { f l , f3 . . . . , / 2 | - 1 ; ~)c(.+1)-1 . . . . , v2,-1 ; e l . . . . . e25-1} ,

C-basis of V. Let d~ �9 Gl.(s, C) be the matr ix of the C-linear t ransformat ion

x-->d(x) of U, relative to (eaj_l}. The centralizer of D in Oh(n + 1)

"i) is then the collection of g �9 Ut(e + t, C) such tha t g = g4 in the 0

C-basis ft. of V, with g~dc=d~g ~. The condition g �9 U*(s + t , C) is expressed g~ = ~ - ~ , g~ �9 U(t -- s) ordinary uni ta ry group, 2((g~)~, (ga)~) + + 2 ((g~),, (g~)~) = ((g~)~, (g~)~) and 2 ((g~)~, (g~)~) - - - - - ((g~),, (g~)~) where (g~)~ is the v ~ row of g . and ( , ) is the ordinary hermit ian pairing of complex

130 Jos~r~ A. Wo~

s t - - s s

m-tuples. We take any x ----- Xajv~r 1 + ~V'bjv2{s+j)_l -Jr .~.ctV2{s+t+:o_ 1 in Q~, 1 1 1

which is given by Qt(v, v) - - 1, and wish to find g ~ Oh(n -~ 1) which centralizes D and gives 9(v2t+x ) ~ x; this last is equivalent to c - a - ~ (gl)1, e + a -~ (ga)l + (ge)l and 2b ~ (g~)l. We m a y choose gl with g~d~ = dctg1-1 and (gl)l = c - - a, set g6 ---- ~1-1, let (g2)~ ~ 2b a n d l e t (g3)1 ~-- c -~ a -- (g6)1. The proof is now identical to tha t of w 9.2, bu t t ha t some condition are conjugate linear.

Now ~ and o~)m (m = 3, 4, 6, 8) consist of homogeneous manifolds of constant positive curvature . For eve ry pair (h, n) of integers with 2h > n > 1, h even and n ~- 1 - - h divisible by 4, there is exact ly one element $~/T)* in ~ . For every quadruple ( h , n ; m ; 2) with h , n and m integers such tha t 2 h > n > 1 , h is even, n ~ - 1 - - h is divisible b y 4 and m ~-- 3, 4, 6 or 8, and where ~ ---- (1, u} is a lat t ice in C with u r R and

exp ( d g ~ / ~ 1/m)2 = X, there is exact ly one element S~a/P.~(u ) of ~f)~.

10. The classification theorem for homogeneous manifolds covered by quadrics

The main result of this paper is

10 .1 . Theorem. Let D be a subgroup o/ Oh(n Jr 1) such that $~h/D is a connected homogeneous mani]old. I] D is ]inite, then is it cyclic or binary polyhedral, and ~ / D is a classical space form, element of ~ , 9 , ~ , ~ , ~) or J . I / D is in/inite but/ully reducible, then it is isomorphic to Z or Z • Z 2 and S~h/D e ~ . I / D is abelian but not/ully reducible, then either it is isomorphic to Z or Z • Z~ with $~h/D E ~ , or it is isomorphic to Z • Z or Z • Z • Z2 with S~h/D ~ o~. I / D is neither abelian nor /ully reducible, then either it is an extension of Z by an element o/order 4 and S'~/D ~ ~oo, or it is an extension Z • Z by an element o/order m, with m ~ 3, 4, 6 or 8, and S~/D ~ ~ m .

Remark. F o r some (h, n), certain families of space forms are excluded. Thus S'~/D~),~ or ~ = if 2h ~ n , i f n + 1 - - h r or i f h is odd; S ~ h / D t ~ D i f 2 h ~ n or if n + l - - h r , ~ h / D ~ f t i f 2h<~n o r i f n + l - - h i sodd; S~h/D~[~ i f 2 h e n + l ; S ~ h / D r o r d ~ ' i f h ~ 0 ( m o d d ) or if n + 1 ~ 0 ( m o d d ) ; S ' ~ h / D ~ i f h is odd or if n + 1 is odd.

Proo/. Let V = R TM, let G be the centralizer of D in Oh(n + 1), and let B be the subalgebra of ~ (V) genera ted b y D. Theorem 2 .5 says tha t G is t ransi t ive on Q~. We will divide the proof into several cases depend-

Homogeneous Manifolds of Constan~ Curvature 131

ing on whether G is irreducible on V, whether D is fully reducible on V and whether B is a division algebra. Note tha t B is a division algebra i f G is irreducible on V, by SCHUR'S Lemma.

10.'2. Suppose t ha t B is a division algebra. We view V as a left vectorspace over B, no te tha t D is a subgroup of the mult ipl icat ive group B*, and (Lemma 8.1) choose a Q-orthonormal R-basis (w~} of V and a s tandard R-basis (a~} of B such t ha t (r --~ dim. B)wrj+~ ~ aiwrj+i. Then s = h/r and m - ~ (n ~ 1)/r are integers, vj = w~(j_l)+l gives us a B-basis (v~} of V,

e and the hermit ian form QB(x, y) : -- ,V, x j y ~ - ~ Z x j y ~ ( x - ~ X x ~ v j and

1 s-i-1 y -~ Z,y~v~ with xj, yj e B) has the proper ty t h a t Q(x, y) -~ <~e(QB(x, y)) and Q~ is given by QB(x, x) = 1.

D is a subgroup of the unimodular group B' of B because D preserves Q~, and D is discrete because i t is properly discontinuous on Q~ ; it follows f rom compactness of B' tha t D is finite. Thus D is a finite subgroup of B ~ acting by left scalar mult ipl icat ion on S~ = {x ~ V : QB (x, x) ~ 1 }. Now S'~/D is a classical space form.

10.3 . Suppose tha t D is fully reducible on V. I f B has an element whose characterist ic polynomial is not a power of an irreducible polynomial, then the rat ional cononical decomposit ion V ---- 27 V, of V by t h a t element is nontri- vial. G preserves each summand V~, so L e m m a 8.3 says tha t 2h -~ n ~- 1 and the decomposit ion is of the form V ---- U O W with U and W maximal to ta l ly isotropic in V. U and W are D-invar iant by L e m m a 8.6, and G is irreducible on each by L e m m a 8 .2 ; it follows tha t the restrictions B] v and B I w are division algebras. These division algebras are each isomorphic to R by Le mm a 8. 4, whence D is scalar on U and on W. We choose an or thonormal basis {vi} of V such t ha t (e~ = v~+~ + v~, ]~ ~ vh+ ~ - - v~ for 1 ~ i ~ h ) {]~} i s a b a s i s o f W and (e i} i s a b a s i s o f U. Then every d e D

is of the form d = '(0 Ia 0)" in the basis {]r e,}. The real numbers a a-l la /

form a discrete subgroup of R* ; Lemma 6.3 then says t h a t D is generated

by some o n e a- l I \ - - /

{/~, e~ ) . Changing to the basis (v i }, we see ~ / D E ~)lf r. Now suppose t ha t every b e B has characterist ic polynomial which is a

power of an irreducible polynomial; we will see t h a t B is a division algebra, whence D is finite, G is irreducible and S~a/D is classical. This is clear i f G is irreducible on V; now assume G reducible. We then have a G-invariant D-invar iant maximal total ly isotropie subspaee U of V b y Lemmas 8.2

9 C ~ H voL 36

1 3 2 J o s E e ~ A . WOLF

and 8.6; the restr ict ion B I v is a division algebra, b y SCHUR's L e m m a , because G is irreducible on U. Let A be the kernel o f the restr ict ion ~ : B - * B I v- B y our hypothes is on character is t ic polynomials , ]l is the set o f all e lements of B whose every eigenvalue is zero; i t follows t h a t ]l is conta ined in the radical of B. B is fully reducible on V, consequence of the full reducibi l i ty of D ; it follows t h a t B is semisimple [1, Th. 4, p. 118]. Thus A ----0, so / : B oo B ] v and B is a division algebra.

10.4. Suppose t h a t D is not fully reducible on V; we will see ,~a/D E ~ t ,

c~ ), ~ . . or c~/)~. F i rs t no te t ha t D is infinite and G is reducible. L e m m a s 8 .2 and 8 .6 give us a G- invar iant D- inva r i an t proper m a x i m a l to ta l ly isotropie subspace U of V, of dimension k -~ n -~ 1 - - h, on which G is irreducible, and gives us a Q-or thonormal R-basis 7 ~ {v~, . . . ,vh+~} such t h a t (e~----vh+ ~ v ~ and ]~-----vh_ ~ . - v j for 1 _~j ~ k ) ( e ~ } is a basis of U. U • is also G- invar ian t and D- invar ian t , and has basis {vk+~, . . . , v n ; e~ . . . . , e k }; i t follows t h a t every e lement of G, D or B has block fo rm

(o al a2 ~- a 5 a a 4

0 a e

in the basis f l~ - { / 1 , ' ' ' , / k ; v k + l , - ' ' , v h ; e ~ , - ' ' , e k } of V. Note t h a t a E G or a c D gives a ~ Oh(n -~ 1), whence a e ~ tal-1 and a 4 E O ( h - - k ) .

Let rl , r 4 and r 6 be the mat r ie representa t ions of respect ive degrees k, h - k and k of G, given b y a - ~ al , a--> a4 and a--> a e. r e is the restric- t ion G--> G 6 ---- G I v , hence irreducible; r 1 is irreducible because i t is contragredient to re; r~ is an or thogonal represen ta t ion of G. Le t a ~ G, ava+ 1 -~

~ - l x j v ~ . A short calculation shows t h a t the first row of a~ is (xa+ 1 - - x 1 . . . . . xh+ ~ - - x ~ ) ; i t follows f rom the t r ans i t iv i ty of G on Q~ t h a t nei ther r~ nor r6 has a nonzero symmet r i c bil inear invar iant .

Le t t~, t4 and t 6 be t he mat r ie representa t ions of respect ive degrees / c , h - - / r a n d /r o f B, g iven b y a - + a l , a - ->a t and a - ->a e, a n d l e t s l , s4 and 8e be the i r restr ict ions to D . te(B ) ---- Be is a division a lgebra b y SCHV~'s L e m m a because it centralizes the irreducible group G e ; thus 8 e is ful ly reducible, sl is fully reducible because i t is eont ragredient to s e, and s 4 is ful ly reducible because i t is an or thogonal representa t ion . E v e r y e lement of D has character is t ic polynomial which is a power of an irreducible po lynomia l : i f not , L e m m a 8.3 would give h ~ k and show tha t we could also assume W - ~ (/~} to be G- invar ian t and D- inva r i an t ; i t would follow t h a t D - ~

(s 1 O s e ) ( D ) so D would be ful ly reducible on V. Thus s 1,s4 and s 6 have the s ame kernel A a n d every d c A is of the form


d = I re lat ive to ~ . 0

s e represents b y unimodular e lements of the division algebra Be, for se(d ) was jus t seen to have the same characteris t ic equat ion as its cont ragredient ts 6 (d) -1, for every d e D . Now s e is self-contragredient , hence equivalent to s 1. I t follows t h a t t 1 and t e are equivalent , hence have the same kernel A. We have seen t h a t there is no nontr iv ia l G-invar iant direct sum decomposi t ion of V, so the character is t ic po lynomia l of a n y e lement of B is a power of an irreducible polynomial . Thus every e lement of the kernel Ker . t 4 has all character is t ic roots zero, hence lies in A because ts(S ) is a division algebra. On the o ther hand, t~(A) is easily seen to be the radical of t4(B), and t4(B )

is semisimple because t4 is fully reducible [1, Th. 4, p. 118], consequence of the fact t h a t s 4 is fully reducible; it follows t h a t ,4 ~ Ker. t4 unless h - - / c

(i ~ -----0. N o w we see t h a t eve ry a ~ N is of the form a = 0 , so ,4

0 is a n i lpotent ideal of B and the difference algebra (see [1], p. 2"/) B - - - ,4 ~ ~(B) , being a real division algebra, is separable [1, Th. 21, p. 44]. This shows t h a t ,4 is the radical of B and t h a t [1, Th. 23, p. 47] B has a subalgebra Be wi th ~, : Be ~ ~e(B). By L e m m a 8.2, we m a y t ake a s t andard R-basis {o~} of By and assume tha t v~+~ = ~ v~+l(r = dim. Be) and t h a t the respect ive R-linear spans of {[~}, {v~+~ . . . . , v~} and {e~} arc Bv- invar iant . No te t h a t we then have e~+~ = ~ e~+~ and ]~+~ = ~ ],~+~. In

i) part icular , By consists of all ba for b ~ B, and A consists of

all matr ices 0 b e for b . I t follows t h a t G centralizes every 0 0

mat r ix d v ~- d4 and t h a t d 1 ~ de, for d e D . 0 6

Given d ~ D , we note t h a t de ~ O h (n ~- 1) ; thus

d v - l d = I d4-1d~ ~ O h ( n + 1). I t follows t h a t d~-ld2 = 0

~- 2 ( t (d~ -~ds ) ) . Set d' -~ d~-~dz. Now let g ~ G . g ( d v - l d ) : (de -~d)g gives us gld ' : d'g~ and g4(2td ') : (2td')g6, whence (d ' . td ' )g, : d'g4 td' -~

-~ gl (d ' . td ') . As g~ --= :g6 -~ and as d ' . td ' is symmet r i c , it follows t h a t d ' . td '

is the m a t r i x of a G~-invariant symmet r i c bi l inear fo rm on U. B u t r6 has no nonzero s y m m e t r i c bil inear invar ian t ; thus d ' . t d / = 0. This gives d~-~d~ ~ 0

and d ( ~ d s ~ - - O , whence d~-~ 0 and d s = 0. We conclude t h a t every

134 JOSEPH Ao WOLF

e lement of D is of the fo rm d - d, re la t ive to ft. Notice t h a t 0

some d e D has ds =/= 0 : D would be ful ly reducible if this were not the case. Choose d ~ D wi th d 3 & 0 and note t h a t d g - ~ g d gives tged3g e - ~ d 8

for every g ~ G. Le t F be the centralizer o f r e (G) in ~ ( U ) . F is a division a lgebra which contains t6(B) as a subalgebra. Given u �9 F, we have tg6(d3u)g e ~-dsu for every g e G. Thus d3u is a bi l inear inva r i an t of re. I t follows t h a t d3u is a nonsingular skew mat r ix . This gives dsu -~ -- t(dau ) : -~ - - l u . td~: ~ud3. I n par t icular , given u , v ~ F, we have uv-~d~ -~. ~(uv).d~-~ = d~-Z.~v.d~.ds-Z.~u.d~ ~ - u v , so F is commuta t ive .

Given b e B and g e G, d g - Z : g-Zd and bg-~ gd gives d ~ g , - ~ : tg~d~ and b~g~ = ~ge-lb~; thus d~-~b~.g~ -~ d~-Z.~g,-Z.b8 : g~d~-~b~, so d~-~b~ ~ F; i t follows t h a t b~ ~ d~- F. Le t Ba be the collection of all mat r i ces b s for b e B; we have jus t seen t h a t the l inear space B~ has dimension a t mos t dim. F. Le t A3 be the collection of matr ices d~ for d ' � 9 A . Given d ' , d o ~ A, we have

0 I / 0 0 I

Thus A is free abehan, isomorphic to the addi t ive group A a. A is discrete, as D is discrete, so A 3 is a lat t ice in Ba. This shows A free abel ian on a t mos t dim. F generators . I f t6(B) ~ R and A = {I}, then se: D ~ s6 (D)cR ' , contradic t ing the hypothes is t h a t D be infinite; thus A ~= {I} i f ts(B) ~ R. I f t6(B ) ~ C and A---- { I } , t h e n s e : D c ~ se(D ) c C' shows D abelian, so B is abelian; then b 3b~ l l tb~ for ~-bib a for every b , b ' e B, whence b 1 = b' r B. This gives t6(B) co R, con t ra ry to our hypothes is ; thus A :/= {I} if te(B ) ~ C. We conclude t h a t A &- {I} and A is free abel ian on a t mos t dim. F generators .

Suppose t6(B) co R and A is infinite cyclic. L e t d = I 0

genera te A. k is even because 83 is skew and nonsingular . I f s6(D ) -~ {I}, then D---- A, so D ~- ~:(-k) and S~h/D~l~t. I f s6(D) r ( I } , t hen se(D ) ~-

= { q - l } , and we choose d - - - - ED. D : {d, A} and 0 0 - -

d:* �9 A, so - - 2d3 is an in tegral mul t ip le of 88. I f - - 2d3 is an even mult iple of 88, then d 3 = m 8 3 for some integer m and - - I ~ - d 0 m ~ D ; we then have D = {- - I , A} = ~ : ( • and S'~h/D ~ ~ , . I f - - 2d 8 is an odd mult ip le


(_i 0 of ~ s , - - 2 d a - - - - ( 2 m - ~ 1)6, , then d~ ~ = 0 - - I generates

0 . 0 - - I / D ; we then have D = ~ ( - - ) and S'~/De ~ t .

Suppose t6(B)c~ R and A is not infinite cyclic. Then F ~ C and A is free abel ian on two generators d and ~ ' ; (~ ---- ~au for some non-reM u et c.

k is divisible b y 4, for we ident i fy tc wi th C and observe t h a t x - ~ d8 x is a C-linear nonsingular t r ans fo rmat ion of U whose ma t r ix mus t be skew because it is a bil inear invar ian t of the C-linear group re(G ) . I f s6(D ) = {I} then D = A, so D = E ( u , ~ - ) and S'~/D r I f s.(D):/= {I}, then s,(D) = { i I } and, as in the last paragraph , we choose d es,-~( - I ) and observe t h a t d ~ ~ A implies - - 2da = m6a A- m ' ~ for integers m, m ' . I f bo th m and m ' are even, then D = { - - I , A} = ~ ( u , 4-) and S'~a/D ~ ) . I f bo th m and m' are odd, we replace d by an appropr ia te d 6 " d 'v and

- - I 0 --�89 assume m = - - 1 = m ' , d = 0 - - I 0 ) , d 2=- Od'

0 0 --I /

Now D = {~ ,d} and - - d = I 0 ; therefore D = 0 I

~ - s 1 ) / 2 , - - ) and S'~h/D ~ , ~ . I f m is odd and m ' is even, we m a y (/ o?) assume d = 0 - - I ; then D = {O ' ,d} - - - -~ ( �89 - 1 , - ) and

0 0 S~a/D e ~ . Final ly, i f m is even and m ' is odd, then we m a y assume

o a - 0 - I ; then D = {~,d} = s 1 8 9 and S~a/Dr ).

0 0 - - I / Suppose t .(B) _--__ C and A is infinite cyclic. Le t 8 = I

0 generate A. As A is a normal subgroup of D , we have d t d -1 c A for every d e D . Using d3d1-1 : tdl-ld3 : did3 and ~3dl -~ : t d l -~ 3 : dld, , this gives dlS~z : mdt 3 for some integer m. Similarly, d-l~d e A gives dl-2~s : P~s for some integer p ; 03 ----- dl-2d12~s = mpds gives p-----m -1, whence m = 4- 1. m = 1 givex d 1 = 4 - I ; m - - - - - - 1 gives di s = - - I . There exists d e D with di s = - - I , for s6(D ) :/: {-4-1}. Given such an e lement

d, we note t h a t d s = d~ = - - I , so the suba lgebra o f \ 0 0 d~ s /

B genera ted b y d is m a p p e d isomorphieal ly on to t6(B) b y t6. We may ,

then, chose d ~ D with d l Z = - I and asume d = d4 . N o w 0

136 Jos~r~ A. Wo~F

D = {d, ~}, d 2 = - I , d ~ d -1 = ~-1, the a r g u m e n t of the last p a r a g r a p h shows k divisible b y 4, and h is even because By ~ C. Thus D = ~ * and 5~h/D ~ ~ .

Suppose t6(B ) ~ C and A is no t infinite cyclic. Then h is even and, looking a t A, k is divisible b y 4. Le t (~, d t generate A. Given d ~ D , dl 2

d1-2 is a real scalar mat r ix ; thus (d6d -1) (d-lt~d) ----- I ~ A 0 I /

shows dl ~ + dl -~ integral . Iden t i fy ing t6 (B) wi th C, set dl = cos (t) +

+ ~ l s i n ( t ) ; d l 2 + d l -~ is then 2 c o s ( 2 t ) , so cos (2 0 = - 1 , - - � 8 9 1 8 9 or 1.

I f c o s ( 2 t ) = - 1, then t---= ~ z t / 2 and d12--- - - - - I ; i t follows t h a t d 2 = - - I . I f cos (2t) = 0, then t - - : t :g /4 or 4- 5zt14 and d l 4 = - - I ; i t follows t h a t d 4 = - - I . I f cos (2 0 = 1, t h e n d 1 = • I . I f cos (2t) ~ - - � 8 9 then t---- ~ :27t /3 or ~ g / 3 , and one easi ly checks dl 2 A - I A - d l - * ~ 0; i t follows t h a t oP ~-- ~ I . Finally, cos (2t) - - �89 is impossible because it would

0 imply t = - t -~ /6 or ~ 7~/6, whence d 2 = d, ~ and

\ 0 0 d l 2 / 0 d 6 : 0 - - I commute ; this would give died3 : d3dl 2,

0 0 - - I / hence d 3 ~ 0, and tell us t h a t A = I because d ~ sCls6(d). Thus ss(D ) is cyclic of order 3, 4, 6 or 8 (for se(D) ~ R'), and we have an e lement d* c D such t h a t s6 : {d* } ~ s6(D). The suba lgebra of B genera ted b y d* maps isomorphical ly onto te(B) under te, so we m a y assume By chosen wi th d* ~ By. This gives d* 3 = 0 for our choice of d*. Now choose J ~ t6(B) with j 2 = _ I , j ~ + l ~ B v with t ~ ( J ~ + l ) : J ; we m a y assume d * : ---- cos (2zt/m)I,+i ~- sin (2g/m)J~+ 1 where m = 3, 4, 6 or 8. d* now acts on A b y (d*dd*-l)a = (cos (4~t/m)I-~ sin (4~ /m)J )d a. Thus D = s where O'~ = u(~a, u : a I -4- b J , and 5~a/D ~ B,~. Q. E. D.

10.5. Corollary. Let D be a subgroup o/ Oa(n -f- 1) such that S~a/D is a connected homogenous manifold. I~ 2h < n or n -4- 1 -- h is odd, then S'~/D is a classical space/orm, element o] ~ , ~ , (~, c~, ~ , or d~'.

For none of the except ional families of space forms occur in these signatures.

Thus :

10.6. Corollary. Let M n be a connected Rzg~r,~v~an homogeneous mani/old el constant negative curvature. Then M'* is isometric to the hyperbolic space

I q . = I-~o.

10.7. Corollary. Let M n be a connected Rr~c~A~-~cian homogeneous manilold


of constant positive curvature. Then M n is isometric to S'*/D where 3n is the sphere II x It ~ k-�89 in a left hermitian (positive definite) vectorspace over a real division algebra t=, D is a finite subgroup of t=', and M n has curvature k. All these RIEMANNian manifolds $n/D are homogeneous o/ constant positive curvature. The only cases are (1) D -~ {I} or Z2, (2) n ~ 1 (mod 2) and D ~ Z q ( q : > 2 ) and (3) n - - 3 ( m o d 4 ) and D ~ * , ~ * , ~ * o r ~ * .

Corollaries 10.6 and 10.7 follow immediately, and give the classification of the RIEMA~Nian homogeneous manifolds of constant nonzero curvature. That classification is known [6].

11. The difficult signature

S~ and S~ are essentially different for h ~-- n -- 1 ; ~1(S~-1) ~-- Z. We will relate the homogeneous manifolds M~_ 1 of constant positive curvature with the homogeneous manifolds S~,_i/D covered by quadrics. The trick is to use the universal covering group of the non-connected group On-l(n ~ 1).

~'* S'~,_1 be the universal pseudo-RI]~MANNian 11.1. Lemma. Let ~ : S,_1 --->

covering, let ~)~-l(n ~ 1) be the full group of isometrics of 5"~n-i, and let D,, be the group of deck transformations o/the covering. Then D,, is a central subgroup

of the identity component SOn-l(n ~- 1) o/ ()~-l(n + 1), D,, is normal in

()n-l(n ~- 1), and we have an epimorphism f: O"-l(n + 1)-> 0"- l(n ~- 1)

o/kernel D,, defined by f(g).~(x) = ~(gx).

Proof. Let G be the normalizer of D, in ( ~ - l ( n - ~ 1), let ~(p)----p,

let ~: be the isotropy subgroup of ()~-l(n + 1) at p, and let K be the

isotropy subgroup of 0 ~-1 (n + 1) at p. (~n-1 (n ~- 1) = G. K because homo-

geneity of S~_i implies that G is transitive on ~ - 1 - f is well defined on G, and ](G) ~ O'~-l(n ~- 1) because every isometry can be lifted; it follows that

](G ,~ K) ~ K. Isomorphic to the linear isotropy group of G at p, G ~ ~: is isomorphic to a subgroup of O~-~(n); K__~ On-~(n) now gives

G ~ K ~ 0 ~-~ (n); this gives /~ c G because /~ is isomorphic to a subgroup

of O~-l(n). Thus (~ - l (n ~ 1) ~- G.-~ ~-- G. I t follows that D. is normal

in O~-l(n ~- 1) and / is well defined on On-l(n q- 1). I t is clear that

S~O n-~ (n ~ l) centralizes D~, being a connected group which normalizes the discrete group D. , and D. is the kernel of f by construction.

Now we need only show D. c S-0 n-1 (n ~ 1); as f is onto, it suffices to

show that On-l(n W 1) and On-l(n ~ 1) have the same finite number of

components. _~ ~ S-O ~-~ (n ~ 1) is connected because 5"~-1 is simply connee-

138 JOSEPH A. WOLF

ted, and ~7 meets every component of On-i(n + 1) because ~ - 1 is connected;

it follows that O~-X (n + 1) and /~ have the same number of components.

oo On-l(n) ~ K, whence /~ and K each has 4 components. On-i(n + 1) has 4 components. Q . E . D .

11.2. We check that g - i ( { 4- I}) is infinite cyclic. This is the case m = 2 of

Lemma. Let S~,_I/Z,~ be homogeneous. Then the fundamental group ~ (S~_I/Z,~) is infinite cyclic.

n--1

Proo/. S'~,_ 1 i sg ivenby xn~+ x~+l~ = 1 + 2:x~ 2, s o t h e m a p s: [0,1]-~5~,_ 1 1

given by s (t) = (0 . . . . , 0 , sin (2gt), cos (2~t)) represents a generator of ~ i ( ~ - 1 ) . Le t b : S~n_~ -> $~n_l/Z,, be the projection and set B = b.~l(5~n_l). We may assume Z,~ generated by I if m - - - - 1 , - - I if m ~ 2 , and

R(1/m) . if m > 2, where It(q) = -- sin (2~q) cos (2~q)]" I t( l /m)

Let v : [0,1] -+ S~_l/Z ~ by v(t) = b (s(t/m)) ; b.s ----- v" ~ ~i(S'~_l/Z,~). Thus B has index m in the cyclic subgroup {v} of ~i($~_i/Z,,). The covering b has multiplicity m, whence B has index m in gi(5~_l/Z~); thus (v} is all of ~i($'~_l/Z,~).

11.3. Certain abelian subgroups of ( ) ' - l (n + 1)

conjugates of every group D such that S'~_I/D describe these groups.

R(s) I f n is odd, every rotation matrix R(s)n+l = "..

30'~-1(n + 1) where R(s) = ( cos (2~s) sin (2~s)~ ' - - sin (2~s) cos (2~s)] "

Q . E . D .

will be seen to contain

is homogeneous. We will

R(s ) ) lies in

The collection

AR of all these rotation matrices is a circle subgroup of SO'~-l(n + 1). I t

follows from Lemma 11.2 that the inverse image AR ~ g-I(AR) is a closed

non-periodic 1-parameter subgroup of SO "-i(n + 1). I f n > 2, we take d to be any nonsingular skew 2 • 2 matrix, and recall

(w 9.2) tha t each translation matrix tH(sd) = I,~_ s 0 sd 0 1 2 + s d

( s r lies in SOn-i(n + 1). The collection AT of all such • tx(sd) is a closed subgroup of $O'~-l(n + 1) isomorphic to Zz • Ri ; Lemmas 11.1 and

11.2 inform us that the inverse image A~ = ~-1 (A~) is a closed subgroup of

~)~-i(n + 1) isomorphic to Z • R i, where the Z is ~ - i ( ( 4 - I } ) and the R 1 is the lifting of the 1-parameter group {tR(sd)}.


/cosh (s) I , sinh (s) Is~ Each hyperbolic rotation Rh(s) = \sinh (s)I~ cosh (s)I2/ lies in S0~(4).

As with the translation matrices, AH = ~-1({ 4- Rh (s)}) is a closed subgroup

of S~02(4) isomorphic to Z • R i, where the Z is g l - i ( { • I}) and the R i is the lifting of the 1-parameter group {Rh(s)}.

Finally, set Az = ~ - 1 ( { i I}) c C)~-'(n -+- 1), and A z = {4- I } .

Lemma. Let D be a subgroup o/ ~)'~-*(n + 1). Then 5"~,_1/D is a homogeneous mani/old i/ and only i/ D is conjugate to a discrete subgroup o/ ~t z, AR, A~, or AH.

Proo/. Suppose ;" S•_I/D homogeneous. Let G' be the centralizer of D in

C)~-~(n-{- 1). The identity component of (~' is transitive on 5"~n-1, and

centralizes D~ by Lemma 11.1 ; it follows that the centralizer G of D . D .

is transitive on S" Thus the centralizer G = / ( G ) of /(D) in O't-l(n + 1) n--l" is transitive on 5"~,_~. We now look at the proof of Theorem 10.1 ; G acts on V = R n+i and B is the subalgebra of ~ ( V ) generated by /(D). I f B is a division algebra then, as in w 10.2, [(D) is conjugate to a subgroup of Az (for n even)or Aa (for n odd). I f /(D) is fully reducible on V and B is not a division algebra then, as in w 10.3, B has an element whose characteristic polynomial is not a power of an irreducible polynomial, n = 3, and /(D) is conjugate to a subgroup of A~. I f /(D) is not fully reducible on V, then, as in w 10.4, /(D) is conjugate to a subgroup of A~,. Thus D is conjugate

to a subgroup of Az , AR, A s or AT. Free and properly discontinuous on

~_~, D is discrete.

Let D be a discrete subgroup of Az, AR, -dH or AT. The centralizer G of ](D) in O~-l(n Jr 1) contains the centralizer of Az , AR, AH or A t , and

is thus transitive on S,n_l ; it follows that the centralizer of D in ~)n-i(n + 1)

is transitive on ~,-1. As D acts effectively, this shows D free on ~ - 1 .

As D is discrete, it easily follows that S~_I/D is a homogeneous manifold. Q . E . D .

11.4. The discrete subgroups of z~z, Aa, AH and z~w are easily described.

Any subgroup of Az is discrete and infinite cyclic. The discrete subgroups

of Aa are the subgroups on one generator; they are discrete. Viewing AH

and AT as closed subgroups Z • R 1 of a vector group R 2, we see that a discrete subgroup is on 1 or 2 generators; a subgroup on 1 generator is discrete and infinite cyclic; one easily checks that a non-cyclic subgroup on 2 generators

is discrete if and only if it does not lie in the identity component of AH (or A~,).

140 JOSEPH A. WoL~

12. The classification theorem for homogeneous manifolds of constant non-zero curvature

Theorem. Let M~ be a connected homogeneous pseudo-RI~,MAN~ian mani/old o/constant positive curvature. I f h < n -- 1, then M~ is isometric to a classical or exceptional space /orm ~/D, element of a/;, ~, (~, d~, (?), ~7,

or It h = n -- tun is isometri

to a mani/old ~ _ i / D , where D is a discrete subgroup o/ a closed subgroup

A z , A , , A ~ or-/~T o/ On-l(n ~-1). I / h : n , then M~ ks isometric to ~ , a component o/ ~ .

Proo/. This is an immediate consequence of Theorem 5, Theorem 10.1 and Lemma 11.3.

Chapter IH. Isotropic and symmetric manifolds of constant curvature

13. The notion of a symmetric or isotropic manifold

Let Q be the metric on a pseudo-RI~xN~ian manifold M~ and, given p E M~, let I(M~)~ be the isotropy subgroup at p of the group I(M~) of isometries. A symmetry at p is an element s~ E/(M~) of order 2 with p as isolated fixed point; such an element is unique and central in /(M~)~ because the tangent map at p is -- I . M~ is symmetric if there is a symmetry at every point. M~ is isotropic if, given p c M~ and a ~ R,/(M~,)~ is transitive on the set of tangentvectors X at p such that Q~ (X, X) ~ a. M~ is strongly isotropic if, given p E M~, ] (M~)~ is transitive on the Q~-orthonormal frames a t p, i.e., !(MD ~ O h(n).

Note that a strongly isotropic manifold is an isotropic manifold of constant curvature; we will prove the converse. We'll also see that an isotropic manifold of constant curvature is symmetric.

I t is well known that a connected manifold is homogeneous if it is either isotropic or symmetric. For, given a tangentvector X at a point p, we have an element sx c ](M~)~, such tha t (sx). X ~ -- X . Now let x, y E M] and join them by a broken geodesic. I f x 1 . . . . . x~ are the successive midpoints of the geodesic segments of this broken geodesic, and if X 1 , . . . , X,~ are the respective tangentvectors to the geodesic segments at x l , . . . , x~, then s~,~sz,~_l . . . sxl(x) ~ y.

Homogeneous M a n i f o l d s o f C o n s t a n t C u r v a t u r e 141

14. The classification theorem for symmetric manifolds o~ zero curvature and for nonholonomie manifolds

14.1. Theorem. Let M'~ be a connected pseudo-RI~MAN~ian mani/old. Then these are equivalent:

1. M'~ is a symmetric mani/old o/ constant curvature zero.

2. M~ is a complete manifold with trivial holonomy group, i.e., parallel translation in M~ is independent o/ path.

3. M'~ is isometric to a mani/old R'~/D where D is a discrete subgroup o/ the underlying vector group R ~ o/ R~.

Proo/. Suppose first that M~ is complete with trivial holonomy. Then M~ has constant curvature zero and admits R~ as universal pseudo-RIEMA~Nian covering manifold. Let D be the group of deck transformations of the universal p s e u d o - R I E ~ i a n covering ~ :R~-+ M~. D is a discrete subgroup of I(R~)-~NO~(n); we write every d eD in the form d-~ (Rg, td) with R~EOa(n) , tdeR'~ and d : x - ~ R ~ ( x ) ~ t d . { R ~ : d e D } is isomorphic to the holonomy group of M~ at ~(0), whence D is a group of pure translations. Thus (2) implies (3). The last part of our argument shows that (3) implies (2).

To see that (3) implies (1), we note that R'~/D carries the structure of a connected abelian LI~ group, and that y--> x2y -~ is the symmetry at x because y-~ y-~ is the symmetry at the coset D.

Now suppose M~ is symmetric of constant curvature zero, let D be the group of deck transformations of the universal pseudo-RIWM~ian covering ~:R~-~M'~ , and write every d E D in the form (Rg, td) as above. Let

n h Q(x, y) -~ - Ex~y~ + Z x j y j on R ~, so Oh(n) C I(R~) is the orthogonal

1 h-t-1 group of Q. Choose d ~ D ; we must show tha t d is a pure translation. The space N'~ : R~h/(d } ({d} is the subgroup of D generated by d) is symmetric because it covers M~ and we can lift each symmetry; it follows that the centralizer G of d in NO a (n) is transitive on R~, and that every symmetry of R~ normalizes (d}.

Given xeR~, choose g e G with g(x)----0. Q (d (x) -- x, d (x) -- x) -~ = Q (gd(x) -- g(x) ,gd(x) -- g(x)) ~-- Q(t~.td). Thus Q ((Ra-- I) x , (Rd-- I) x ) § ~-2Q((R d - - I ) x , t ~ ) ~ 0 for every x E R~. Replacing x by u x as u runs through R, we see that Q((R~ -- I ) x , (R~ -- I ) x ) ~-- 0 : Q((R~ -- I )x , t~) for every x ~ R~. Polarizing the first equality, then expanding and using R~cOa(n), we get Q ( x , ( 2 1 - - R ~ - - R ~ - ~ ) y ) - ~ 0 for x , y ~ R ~ . Thus 2 1 ~ R ~ - R ~ -1, whence ( R d - - I ) 2~-0 . We conclude that R~----I+~V with N ~ - 0, N(R~) is totally Q-isotropie, and t~ is Q-orthogonal to N(R~).

142 J o s ~ H A. W o I ~

In part icular , (Ra)" -~ I Jr raN , whence ei ther R a ---- I or R a has infinite order. Thus {d } is ei ther tr ivial or infinite cyclic.

We now use the fact t h a t {d} is normalized by the s y m m e t r y s o ---- ( -- I , 0) of R~ a t 0. sodso -1 ~ (Ra, -- ta) is ei ther d or d -1, as {d} is either tr ivial or infinite cyclic, sods~ -1 :/= d as td ~= 0, for d has no fixed point. Thus sodso -1 = d -1 ~ (Rd -1, - - Ra- l ta) . In part icular, R d 2 ---- I . As R a 2 ~ -

I 4- 2(Rd -- I ) , as seen in the last paragraph, we conclude tha t Rd -- I and d is a pure translat ion. Thus (1) implies (3). Q . E . D .

14.2. C~)rollary. Let M n be a connected R x E M A ~ i a n manifold. Then these are equivalent:

1. M'* is a symmetric manifold of constant curvature zero.

2. M'* is a complete manifold with trivial holonomy group.

3. M n is isometric to a product of a euclidean space with a flat torus, i. e., M n is isometric to a quotient R ' / D of a (positive-definite) euclidean space by a discrete vector subgroup.

4. M n is a homogeneous manifold of constant curvature zero.

Proof. The theorem establishes the equivalence of (1), (2) and (3), and (3) tr ivial ly implies (4). Now assume M n homogeneous of constant zero curvature , and let d ~- (Rd, ta) be a deck t ransformat ion of the universal R I E ~ _ ~ i a n covering ~ : Rn--> M ~. In the proof t ha t (1) imply (3) in the theorem, we saw tha t (R d - - I ) (R '*) must be to ta l ly isotropic. Thus R a - ~ I , for R n is a positive definite euclidean space. Q . E . D .

15. The classification theorem for isotropie manifolds of zero curvature

Theorem. Let M~ be a connected p s e u d o - R I E ~ N i a n manifold of constant curvature zero. Then these are equivalent:

1. M~ is isotropic.

2. M~ is strongly isotropic.

3. M~ is isometric to R~.

Proof. I t is clear t ha t (3) implies (2) and t ha t (2) implies (1); now assume M~ isotropie. Le t D be the group of deck t ransformat ions of the universal pseudo- R ~ M A ~ i a n covering ~ : R~-->M~, and wri te d ---- (R~, ta) eNOh(n) for every d �9 D . The centralizer G of D in NOa(n) is t ransi t ive on R~, and the i so t ropy subgroup K of G a t 0 is irreducible when we view K as a group of linear t ransformat ions of the vectorspace V ----- R~. As in the proof of the last theorem, t rans i t iv i ty of G implies t h a t every ( R ~ - 1 ) 2 ---~ O,


whence R ~ - I is nilpotent. K centralizes every R d - - I by construction, whence R~ -- I ---- 0 by Sc~vR'S Lemma. Thus D is a group of pure translations (I, t~). Let W be the subspace of V spanned by the vectors t~ with d e D. Identify V with the tangentspace to R~ at 0 ; then a tangentvector X E V is an element of W if and only if the geodesic exp (t~. X) in M~ lies in a compact set; it follows that W is K-invariant. Thus W ---- 0 or W ---- V, by the irreducibility of K. W :fi V because K cannot carry a closed geodesic of M~ onto an open geodesic. We conclude that D consists only of (I, 0).

Q.E.D.

16. The classification theorem for symmetric and isotropie manifolds of constant nonzero curvature

When considering pseudo-RI~ANNian symmetric, isotropic or strongly isotropic manifolds of constant nonzero curvature, we may assume that

curvature positive; for I (S~/D) : / (~ t~_h /n ) .

16.1. Theorem. Let M~ be a connected pseudo-RI]~MA~Nian mani]old o/ constant positive curvature. Then these are equivalent:


2. M~ is isotropic.

3. M~ is symmetric.

4. M~ is complete, so we can speak o/ the group D o/ deck transformations o/ 7,, '~" /urthermore, D is a the universal pseudo-RIv.~ANNian covering fl : Sa -* Ma ,

normal subgroup o/ ] (S~).

5. I / h < n -- 1, then M~ is isometric to S~a or S~a/{• I ) , element o/

if h - ~ n , then M~ is isometric to ~ , element o/ ~ if h : n - - 1 , and if

A z denotes the complete inverse image o/ (Jr I ) c O'~-l(n ~- 1) under the

epimorphism / : ~ - 1 (n + 1) -~ 0 '~-1 (n ~- 1) o/ Lemma 1 I. 1, then M~ is

8,_1/D where D is a subgroup o/ the in/inite cyclic isometric to a mani/old ~n

group A z . 6. 5~h/ { ~- I ) admits a pseudo-RIE~L~NNian covering by M~.

16.2. Proof. If M~ satisfies any of the 6 conditions, then it is homogeneous. I f n ----- h, this imphes that M~ is isometric to 5~/{-t- I ), which is isometric

to 5"~n, whence M~ satisfies all 6 conditions. Now we will assume h ~ n. Suppose first that M~ admits a pseudo-RIEM~m~ian covering a : 5"~a--~ M~ (5~h is connected because h < n), and let Da be the group of deck transformations of the covering. ~ is the universal covering, hence normal, if h < n -- 1.

144 Jos,~PH A. W o ~

is also normal if h -~ n -- 1, for, in the terminology of Lemma 11.1 and of our various conditions, fl ~ ~ .~ and we need only check that D~ ~ ](D) ; the latter is clear because f-l(D~) is D, being the normalizer of D~ in D. Thus M~ = ~/D~, . Then, as every discrete normal subgroup of Oh(n + 1) is contained in { t I} , conditions (4), (5) and (6) are each equivalent to the condition

7. D~ c {4- I} .

As (7) implies (1), and (l) implies (2) and (3), we need only check that (2) and (3) each implies (7).

16.3. Suppose that M~ ~ $'~/D~ is symmetric. Let b c D~ and let B be the subgroup of D e generated by b; the manifold N~ = $'~/B covers M~, so every symmetry of M~ lifts to N~. Thus N] is symmetric. Now every symmetry of N~ lifts to a symmetry of S~, whence every symmetry of S~ normalizes B. Thus the symmetry s ~ diag. {-- 1, -- 1 , . . . , -- 1, 1} to 3~ at Po = (0 . . . . ,0 , 1) normalizes every conjugate of B. We now use homogeneity of M~ and the list provided byTheorem 10.1 to prove Da c { 4- I}, which is the same as M~ ~ ~ . I f M~ is classical but not contained in ~ , then we have an element b eD~ which is conjugate to b ' -~ diag. {R(v) . . . . . R(v)}

where R(v) = ( cos (2~v) sin(2~v)~ - - sin (2~v) cos (2~v)] and v is a rational number such

\

that sin ( 2 ~ v ) r 0; s normalizes the group B' generated by b ~, whence sb' s - lb ~-1 ~ Br ; this contradicts sin (2~v) ~ 0 because sb' s- lb '-1 -~

-----diag. {I 2 . . . . . I~, S} where I 2 ~ (~ 1 ) a n d

S - - - - ( - - 0

I f M~ ~g/o, then we have an element b eD~ which is conjugate to b ' -= /cosh(v) sinh(v)~

= 4- diag. { R h ( v ) , . . . , R h ( v ) ) where Rh(v ) = ~sinh(v) cosh(v)/ and

sinh (v) r 0 ; the group B' generated by b ~ is infinite cyclic and normalized by 8, whence 8bts-~b '-x -~ I or b~-*; this contradicts sinh(v):~ 0 because sb' 8-1b '-x = diag. {I~ . . . . , I~, S ' } where

1 ~ ) . R h ( - - v ) . S'---- ( - - 0

Now suppose that M~ is neither classicM nor in ~ ; by Theorem 10.1 and the homogeneity of M~, D e has an element b which is conjugate to some tH(d) wi th d skew, nonsingular and of degree k ~ n ~ 1 - - h ~ h , I f


{vi} is our given O~(n + 1)-orthonormal basis of V = R "+l, we may set e~ = vn+ ~ + v~ and /i = v~+~ -- vi, and

tn (d) = Ia_ , in the basis 0

fly = { h , - . . , / ~ ; v k + ~ , . . . , vh ; ea . . . . , e~} o f V. As p0 = v~+~,

s = -- Ih_ ~ in the basis fir, 0

where A-----diag. {-- 1 . . . . , - - 1 ,0} and E = I ~ + A . The group B ' generated by tH (d) is infinite cyclic and normalized by s, whence s. tH (d). s -1 is tH(:k d). Using A 2 + E 2 = I~ and A E = 0 = E A , we see tha t

( ! , 0 2 A d A I s. t , (d). s -1 = Ia_~O O R l-

whence A dA ~ -4-d. _As A is singular and d is nonsingular, this is impossible. Thus (3) implies (7).

16.4. Suppose tha t M~ -~ S~/Da is isotropic, and let b c D , . Let G be the normalizer of Da in 0 a (n + 1) and let H be the isotropy subgroup a t x ~ S~. H is transit ive on the tangentvectors to S~ of any given length, whence H(y) is uncountable if + x : ~ y r I f b(x) ~ i x , this shows tha t {b-lgbg-i(x) : ff e H} is uncountable. On the other hand, b-lgbg -I ~ D~ for f f E H , and Da is countable. Thus b ( x ) = 4 - x for every b~D~ and every x c S~a. This shows tha t every b r Da is • I , whence (2) implies (7).

16.5. We have now seen tha t our 6 conditions are equivalent if M~ admits a pseudo-RIEMiN~ian covering by S~a. Now we may assume h = n - 1. We retain the notat ion of L e m m a 11.1. The / - image of a discrete normal sub-

group of ()n-i(n + 1) is a countable normal subgroup of On-l(n + 1), thus contained in ( + I } ; it follows tha t (4), (5) and (6) are each equivalent to

8. D c A, , i .e . , / ( n ) c {:k 1} . As (8) implies (1), and (1) implies (2) and (3), the theorem will be proved when we check tha t (2) and (3) each implies (8), when h = n -- 1.

16.6. Le t M~_~ be symmetric. As every symmet ry of Mnn_l lifts to a sym-

met ry of 5"~n-1, we see tha t every symmet ry of 5"~n-1 normalizes D ; as every

symmet ry of S~n_l lifts to a symmet ry of ~ - 1 , it follows tha t every symmet ry of S~,_1 normalizes / (D) . _As in w 16.3, we see /(D) c {q-1}. Thus (3) implies (8).

146 Jos~P~ A. WoT.]r

Let M~_ 1 be isotropic, and let G be the normalizer of D in ~n-l(n Jr 1). Every isotropy subgroup of G is transitive on the tangentvectors, at the rele- vant point, of any given length; it follows that the normalizer of f(D) in Oa(n ~-1) is f(G) and has the same property. As in w 16.4, we see /(D) c {:t: I}. Thus (2) implies (8). Q . E . D .

17. Applications

17.1. Theorem. Let M~ be a connected isotropic pse~o-RIBM~NNian manifold of constant curvature. Then M~ is symmetric.

Proof. This is an immediate consequence of Theorems 14.1, 15 and 16.1.

17.2. Theorem. Let M'~ be a pseudo-RIEM~v~ian manifold. Then these are equivalent:


2. M~ is an isotropic manifold of constant curvature.

Proof. Note that we may assume M~ connected. The theorem is then an immediate consequence of Theorems 15 and 16.1, and of the fact that a strongly isotropic manifold has constant curvature. Q . E . D .

17.3. Theorem. Let ~ : N'~-+ M~ be a pseudo-RxEM~N~ian covering with N'~ strongly isotropic, and let D be the group of deck transformations of the covering. Then M~ is strongly isotropic of and only if D is a normal subgroup of I(N'~) ; in that case, the covering is normal.

Proof. N~ has constant curvature, and the theorem follows trivially from Theorem 15 if that curvature is zero; we may now assume that N~ has constant positive curvature. We have pseudo-RIEMANNian coverings

a : 5~a-+ ~ and fl: 5~a--> M~ with respective groups D~ and D~ of deck transforms.

Suppose M~ strongly isotropic. Then Da c D~, Da and D~ are normal

in 1(5"~h ) by Theorem 16.1, and c~ induces an isomorphism I(S~a)/D~--+I(N~) which carries D~ onto D. Thus D is normal in I(N~), and ~ is normal.

Suppose D normal in / (N~). We still have an isomorphism ] (~)/D~ -+ I (N~) because 2~a is strongly isotropic, whence the inverse image D~ of D is

normal in 1(~) . By Theorem 16.1, M~ is strongly isotropic. Q . E . D .

Added in proof: 1. Remark on Corollary 10.5. I f 2h ~ n , then D is finitm even if ~ / D is not assumed to be homogeneous. See [7].


2. Remark on Corollary 14.2. The analogous result is true for manifolds M~ provided that h = 1 or h = n - - 1 (the Lorenz signatures), or that M~ is compact, or that n ~ 4; it is false if these assumptions are not made. See [8].

The Institute for Advanced Study

REFERENCES

[1] A. A. ALBERT, ~.~try.cture o/Algebras. Amer. Math. Soc. Colloquium Publications, vol. 24, 1939. [2] S. HELOASO~, DiHeren$ial operators on homogeneous s~aces. Acta Math., vol. 102 (1960),

239-299. [3] N. Hicxs, A theorem on a/line connexions. Illinois J. Math., vol. 3 (1959), 242-254. [4] K. N o ~ z u , lnvarlant a]]ine connections on homogeneou~ s:oaees. Am. J. Math., vol. 76 (1954),

33-65. [5] W. Tmav.L~,~iz~ und H. S~.I~RT, Topologische Untersuchung... sphd'risehen Raumes. Math.

Ann., vol. 104 (1930), 1-70; ibid. Math. Ann., vol. 107 (1932), 543-586. [6] J . A. WOLF, Sur la classi]ication des vari~t~s ~ L ~ i e n n e ~ homog~nes ~ courbure eoustan$e.

C. r. Acad. Sci. Paris, vol. 250 (1960), 3443-3445.

Added in proof: [7] J . A. WOLF, The Clifford-Klein space forms of indefinite metric. Ann. of Math., to appear. [8] J . A . WOLF, Homogeneous manifolds of zero eurvaturv. Proc. Amer. Math. Soc., to appear.

(Received March 24, 1961)

10 C~H vol. 86

homogeneous manifolds of constant curvature

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