on locally symmetric spaces of non-negative curvature and certain other locally homogeneous spaces

On Locally Symmetric Spaces of Non-negative Curvature and certain other Locally Homogeneous Spaces

by JosEPI~ A. WOLF ~, Pr inceton (N. J .)

To Professor Georges de Rham on his sixtieth b i r thday

1. Introduction and summary

This paper is a s tudy of the global s t ructure of the complete connected locally symmetr ic RIEMAI~Nian manifolds N in which every sectional curvature is non-negative. Our main result is t ha t the fundamenta l group s l (N) is a finite 2-group if the EULER-POINCARI~ characterist ic (singular theory)

(N) r 0. In fact, t ha t result is proved under slightly weaker conditions on N . The first principle result (Theorem 3.1) states t ha t there is a real analyt ic

covering N ' - + N of finite mult ipl ici ty and a real analyt ic deformat ion re t ract ion of N onto a compact to ta l ly geodesic submanifold, such tha t N ~ - - E • T • M ' where E is a EUCLIDEan space, T is a torus, M ' is a compact simply connected RxE~lA~-~cian symmetr ic space, and the deformat ion re t ract ion of N lifts to a deformat ion re t ract ion of N ' onto T • M ' . In particular, the bet t i numbers (singular theory) of N are finite and the EULEI~-PoINCXa~ characterist ic Z(N) is defined. Theorem 3.1 then states tha t z(N)--> 0, and t ha t the fundamenta l group ~I(N) is a finite 2-group if ;~(N) r 0.

The second principle result (Theorem 3.2) gives a general m e th o d of con- s t ruct ing all manifolds 57 with Z (iV) r 0. Application of this me thod is a combinatorial problem which requires a classification (up to global isometry) of the space forms of the irreducible compact s imply connected RIEMANNian symmetr ic manifolds S with Z (S) r 0. Tha t classification problem is solved in w 5. We first prove (Theorem 5.1) t ha t S is equal to any of its space forms unless S is a GRASSMXNN manifold, SO(2n) /U(n) , Sp(n) /U(n) , ET/A ~ or ET/E e �9 T ~. We have a l ready classified the space forms of GI~ASSMANN manifolds of nonzero characterist ic [13]; the result is recalled as Theorem 5.3. We then (Theorems 5.4 - 5.7) classify the space forms of the other possibilities of S. F r o m these classification theorems we are able (Theorem 6.2) to give a necessary and sufficient condition on the set of factors of a product M'

1) The author thanks the National Science Foundation for fellowship support during the period of preparation of this paper. His present address is: Department of Mathematics, University of California, Berkeley 4, California.

JOSEPtr A. WOLF On locally symmetric spaces of non-negative curvaturo 267

of manifolds S, t ha t every space form of M ' have abelian fundamenta l group. I f M ' is irreducible, the condition is au tomat ica l ly fulfilled. Finally, in w 6.5, we give a good descript ion of the possibilities for a manifold N with g (N) r 0 when, in the universal RIEMANNian covering manifold M 0 • M r (M o EUCLIDean and M ' compact ; this is the M ~ which occurs in N ' ) , M ' satisfies the commu- t a t i v i t y conditions of Theorem 6.2.

Some par t s of Theorem 3.1 do not fully use the hypotheses on N . This leads us to define a RxEMa~Nian nilmani/old to be a RIEMANNian manifold which admits a t rans i t ive n i lpotent group of isometrics. We prove (Theorem 4.2) tha t a connected I~IEMA.'~Nian nilmanifold is isometric to a connected ni lpotent Lie group in a left invar ian t RIEMANNian metric, t h a t the nilradical of its connected group of isometrics is the only connected t ransi t ive n i lpotent group of isometries, and t ha t its full group of isometries is the semidirect p roduc t of this nilradical wi th an i so t ropy group. Now let 2 / b e a RIEMANNian manifold wi th universal RIEMANNian covering manifold of the fo rm M 0 • M ' where M 0 is a I~IEMANNian nilmanifold and M ' is a compac t RIEMANNian homogeneous manifold. Theorem 4.1 provides a real analyt ic covering

N ' = E • N" • M ' - ~ N

of some finite mul t ip l ic i ty r > 0, where E is a EucLIDean space and N" is a compac t nilmanifold. While I a m unfor tuna te ly unable to re t rac t N onto a compac t submanifo ld unless M 0 is a EUCLIDean space (and so cannot p rove the be t t i numbers of N to be finite, and so cannot assert t ha t z (N) is defined)

1 it is shown (Proposit ion 4.4) t h a t Z* (N) : r Z (N') is a topological invar ian t

of N . Theorem 4.1 then s ta tes t ha t z*(N) is an integer, t h a t z*(N) ~ O, t h a t z*(N) = z(N) if M o is a EUCLIDean space, t h a t s~(N) is finite i f Z * ( I V ) r and t ha t st(IV) is a finite 2-group if z* (N) =~0 and M ' is RIEMANNian symmet r ic .

The " ra t ional EULEmPoI~rCA~ character is t ic" Z* was invented by C.T.C. WALL [10] in another context . D .B .A . EPsTm~ suggested t ha t I use it here, and gave valuable suggestions for adapt ing it to noncompac t spaces and then proving it well defined.

B y Theorem 3. l, we mean the theorem in w 3.1. Similarly, Theorem 3.9 is the theorem in w 3.9, etc.

Added in proo[: B y different methods, J .C . SANWAL has obta ined the flat case of the f o m t h corollary of w and has shown tha t the fundamenta l group of a complete flag RIS.MANNian manifold is isomorphic to t h a t of a compac t flat RIS.MA~ian manifold, special case of our Theorem 3.1.

268 JOSEPH A. WOLF

2. Preliminaries and notation

We will assume familiari ty with LIE groups and discrete subgroups, RIE- MA~vNian manifolds, and covering spaces.

2 .1 . LIE groups and algebras. I f G is a LIE group, then G O will denote its iden t i ty component, ffi will denote its LIE algebra, exp : ~ -+ G O will be the exponential mapping, and adjoint representation of G on ~ will be denoted by "ad" . I f H is a LIE subgroup of G, then .~ is viewed as a subalgebra of ffi. I f ~ is a subalgebra of N, then the corresponding analytic subgroup o/ G is the analytic (---- connected LIE) subgroup generated by the image of .~ under the exponential mapping of G.

I f G and H are LIE groups and fl : H - + Aut (G) is a continuous homomorphism of H into the group of automorphisms of G, then the semidirect product G. ~H (denoted G. H when there is no possibility of confusion) is the manifold G • H with group structure (gl, hi) (g~, h2) ~- (g~" fl(hl)g2, h~h2). G. H is a LIE group, G and H are closed subgroups under identifications g-+ (g, l) and h-+ (1, h) (we always use 1 to denote the group identity), and G is a normal subgroup. The two extreme cases are when fl is trivial, so G. H is the direct product G • H , and when fl is faithful (trivial kernel), so H may be viewed as a group of automorphisms of G if fl(H) is closed in Aut(G).

The compact classical groups are the orthogonal groups 0(n) in n real variables, the ident i ty components, SO (n), the special ( = determinant 1) orthogonal groups, the un i ta ry groups U (n) in n complex variables and the special un i ta ry groups SU(n), the symplectic groups Sp(n) which are the uni tary groups in n quaternion variables, and the universal covering groups Spin (n) of SO (n). T 'n will denote an m-torus. An, B~, Cn, Dn, Gz, F 4, E G, E7 and E s will refer both to the CARTA~ classification types and to compact connected groups of those types. In boldface, these letters will denote the compact s imply connected groups. For example, A~ = SU(n + 1), B~ = S p i n ( 2 n + l ) , C~ = Sp(n), D n = Spin(2n), and F 4 is the group ofisometries of the CAYLEY elliptic plane.

2. 2. Discrete groups. A subgroup F of a topological group G is called discrete if it is a discrete subset, i.e., if there is a neighborhood U of 1 e G such tha t / ~ U = {1}. /~ is called uni/orm in G if (F denotes the topological closure) G/F is compact.

L e t / ~ be a topological group and let X be a topological space. An action of /1 on X is a homomorphism of /~ into the group of homeomorphisms of X such tha t the associated map _F • X--~ X is continuous. We write 7 (x) for the image of @, x). The action is e//ective if 1 # ? e F implies y(x) # x for

On locally symmetric spaces of non-negative curvature 269

some element x e X ; the action is /ree if 1 c y E F implies 7(x) =/:x for every element x ~ X ; the action is properly discontinuous if every x E X has a neighborhood which meets its t ransforms by only a finite number of elements of F .

Le t /1 and K be subgroups of G, K closed in G. Then there is a na tura l action y : g K - + ~ g K of F on the coset space G/K. I f G is a Lzn group, or even locally compact with only finitely man y components, and if K is compact, then the action is proper ly discontinuous if and only i f /~ is discrete in G. In any case, the action is free if and only if 1 is the only element of F conjugate (in G) to an element of K , and the identification space of G/K under F is the double eoset space F ~ G / K .

2 .3 . Isometrics and product structure. An isometry of a RIEMAN~ian manifold is an auUomorphism of the I~EMAN~ian structure. I f M is a RIEMA~Nian manifold, then its lull group o/(---- group of all) isometrics is a LIE group denoted I ( M ) ; the connected group o/ isometrics is the ident i ty component I(M)0; following tradit ion, we write I0(M ) for I(M)0. M is homogeneous i f I (M) is t ransi t ive on the points of M. I f M is homogeneous and connected, and if x e M, then g-> g (x) induces differentiable homeomorphisms of M with the cosetspaces I(M)/K and Io(M)/(Io(M)~K) where K = {geI(M):g(x)-~ x} is the isotropy subgroup o/ I (M) at x ; K is compact . I f s ~ I (M) has square 1 and has x e M as an isolated fixed point, then s is a symmetry to M at x ; if M is connected, s is unique because it induces - I (I --~ identi ty) on the tangentspace M~. M is symmetric i f it has a sym m et ry at each of its points. I f M is connected and symmetric , then any two points x and y can be joined by a broken geodesic, and the product of the symmetr ies to M at the midpoints of the geodesic segments will send x to y ; thus M is homogeneous.

Let M be complete and simply connected. Then [7] M is isometric to a product M 0 x M~ • . . . x M s where ~/0 is a EUCLIDean space (the EvvLzDean /actor o/ M) and the other Mi, the irreducible/actors o/ M, are irreducible, i.e., are non-EucLIDean and not locally products of lower dimensional manifolds. This ~E RltAM decomposition is unique up to the order of the factors. M is homogeneous (resp. symmetr ic) if and only if each of the M~ is homogeneous (resp. symmetric) . Ident i fy ing M with M 0 X . . . X M~ and lett ing I (Mi) act on M by acting on M~ in the usual way and by acting tr ivial ly on the other Mj, I (M) is generated by the I (Mi) and by all permuta t ions on sets of mutua l ly isometric factors M~. In particular, I0(M)----I0(M0) X

x L (M1) x . . . x L ( M , ) .

2 . 4 . Curvature, characteristic and submanifolds. I f S is a two dimensional subspace of a tangentspace M~ to a RIEMANNian manifold M, then in a neigh-

270 gos~r~ A. Wor~

borhood of x the geodesics of M through x tangent to S form a surface; the sectional curvature o/ M at (S, x) is the G.~vssIA~" curvature of t ha t surface at x. In a E v c ~ D e a n space, every sectional curvature is zero. In a compact RIEM~Nian symmetr ic space, every sectional curvature is ~ 0. In a noncompact irreducible RIEl~A~ian symmetr ic space, every sectional curvature is ~ 0 and some are <: 0. In part icular, if M is a complete simply con- nec ted R I E M x ~ i a n symmetr ic space, then M has every sectional curvature

0 if and only if every irreducible factor of M is compact .

A submanifold of M is totally geodesic i f and only if every geodesic of the submanifold is a geodesic of M, i.e., i f and only if the submanifold contains every geodesic of M to which it is tangent . I f X is a to ta l ly geodesic submanifold of M , x �9 X and S is a two dimensional subspace of X~, then it is clear t ha t M and X have the same sectional curvature a t (S, x). In part icular, the sectional curvatures of X satisfy any bounds satisfied by those of M.

The rank of a compact LIE group is the common dimension of its maximai toral subgroups. I f K is a closed subgroup of a compact LI~. group G, then [8] the EULER-POI~C~]~ characterist ic (in any homology or eohomology theory) y~ (G/K) ~ 0, and ~(G/K) > 0 i f and only if rank. G = rank. K .

2. 5. RIEMANxian coverings and locally symmetric spaces. A RI~MaN~ian covering is a covering 7~ : M --> N of connected RIEMAN~cian manifolds where is a local isometry. I t is then easily seen t ha t the group T 'of deck t ransformat ions of ~ (homeomorphisms y : M - > M with ~r ~ ,~. y) is a discrete subgroup of I (M) acting freely and proper ly discontinuously on M . I f M is simply connected, t h e n / ' is identified with the fundamenta l group :r 1 (N) and N is identified with the quot ient space M/F. Conversely, if M is a connected RI~MANNian manifold and F is a subgroup of I (M) acting freely and proper ly discontinuously, then M/I" admits a unique RIEMA~Nian s t ructure such tha t the project ion M-+ M / F is a RIEM_~-Nian covering.

A RI]~Mxcc~ian manifold M is locally symmetric if every x �9 M has an open neighborhood which, in the induced RIEMA~C~ian structure, admits a s y m m e t r y at x. This is the case if M is symmetr ic , if M is a RIEMXCcNian covering manifold of a locally symmetr ic RInMA~tcian manifold, or if M admits a RI~- Mx~Nian covering by a locally symmetr ic R I ~ . M ~ i a n manifold. M is complete, connected and locally symmetric , if and only if its universal RI~.MA~Nian covering manifold is symmetr ic . In part icular , M is a complete connected locally symmetric RIEMA~sian mani/old with every sectional curvature > O, i/ and only i / the universal R I E M ~ i a n covering mani]old o] M is the product o/ a EtrcLiDean space and a compact simply connected RIEMaN~ian symmetric space. This is the sort of manifold with which we shall concern ourselves here.

On locally s y m m e t r i c spaces of non-nega t ive cu rva tu r e 271

3. The structure theorems lor locally symmetric spaces

Our main results on the s t ructure of locally symmetr ic spaces of non- negative curvature are:

3 .1 . Topological Structure Theorem. Let N be a complete connected locally symmetric RI~Ma~ian mani/old with every sectional curvature ~ O. Then:

1. There is a real analytic covering N'--> N o/ /inite multiplicity where N ' is the product o/ a EvcLxDean space, a torus, and a compact simply connected R I E M . ~ i a n symmetric space. This covering need not be RIEMaN~ian. In particular, the/undamental group ~1 (N) has a/ tee abelian subgroup o/[inite index.

2. There is a real analytic de/ormation retraction o/ N onto a compact totally geodesic submani/old which lilts to a de/ormation retraction o/ N ' onto the product o/i t8 toral and compact simply connected/actors. I n particular the betti numbers o/ 57 are / in i te /or singular homology and cohomology, and the EVLzR-PozzCC~ characteristic z ( N ) , alternating sum o/ the betti numbers, is a well de/ined integer. We have )~ (N) >= O.

3. I / 7, (N) :/: O, then ~1 (N) i8 a/ in i te 2-group (/inite o/ some order 2a). Given the first and second s ta tements above, it is easily seen tha t ~1 (N)

must be finite when Z (N) r 0, bu t it is a bit surprising tha t ~1 (N) must be a 2-group. This comes from an examinat ion of the universal covering of N and the form of the elements of J"~I(N), and from ]~. CA~TA~'S determinat ion [4] of the :full groups of isometrics of symmetr ic spaces :

3 .2 . 6eometr ic Structure Theorem. Let M ~ M o • M~ • . . . • M~ where M o is a EvcizDean space and each M i (i :> O) is a compact connected simply connected irreducible Rx~Ma~Tian symmetric space with Z (M~) :> O. Let Z, be a group o/isometrics acting/reely on M~ • . . . • M~, let / be a homomorphism o/ X into the orthogonal group o/ Mo, and let I ~ be the group o/isometrics o/ M consisting o/ all ~ ~- /((l) • ~. Then F i8 isomorphic to X and is a/ ini te 2-group, and M / F is a complete connected locally symmetric R i~ i~y~ ian mani/old with every sectional curvature ~ 0 and Ew~R-PoINcaR~ characteristic )~ (M/F) :> O. I / a n element o / Z has order 2 ~+1 , then it induces a trans/ormation

(X 1 . . . . . X m ) ---> ( T W i n , X I , . . . , X m - - i )

on a product o/ m distinct mutually isometric/actors M~ o/ M , where either m -: 2 u and v is a / i xed point/ree involutive isometry, or m ~ 2 u-1 and (/or some n ~ 2) each o/ these M i is isometric to the oriented real GJ~ass~zr maul/old SO (4n)/SO (2n) • SO (2n), and ~a is a / i xed point/ree involutive i8ometry.

Conversely, every complete connected locally symmetric R ~ M ~ i a n mani/old, with all curvatures ~ 0 and nonzero characteristic, is isometric to a mani/old M /F described above.

272 JOSEPH A. WOLF

3. 3. Outline of proof. The remainder of w 3 is devoted to proving Theorems 3.1 and 3.2.

We ident i fy ~1 (N) with the group /" of deck t ransformat ions of the universal RIEMA~Nian covering M--> N . To obtain the finite covering and the retract ion of N , we find a free abelian subgroup A of finite index in F and sub- mit M, A and F to various deformations. The existence o f / ! (w 3.4) is due to L.AtIsL2t~D]~R. The deformations of A (w 3.5) are done in sufficient genera l i ty for their applications in w 4 as well as in w 3. I t is then (w 3.8) proved that , if ; / ( N ) r 0, then z (N) > 0, ~I(N) is finite, and the converse of Theorem 3.2 holds; this is done by combining the re t ract ion and the finite covering. I t then suffices to prove tha t 27 is a 2-group whose elements induce the t ransformations given; this is done in w167 3 . 1 0 - 3 . 1 1 , and is based on a theorem (w 3.9) tha t T ~ has a fixed point if ~ is an i sometry of an M~.

3 . 4 . The free abelian subgroup of finite index in ~1 (N) will be exhibi ted as a consequence of a result of L.AusI~A~DE~ ([2], Th. 3) which requires some interpreta t ion. The precise s ta tement , slightly sharpened, is:

Proposition (L.AusLA~DER ([2], Th. 3)). Let D be a discrete subgroup o/a semidirect product H �9 C, where H is a connected simply connected nilpotent L~E group acted upon (by automorphisms, but not necessarily e//ectively) by a compact LIE group C. Then D* -~ D ,-, (DH)o is a subgroup o/]ini te index in D, and D* = A • B where A is a finite abelian group and B is isomorphic to a discrete subgroup with compact quotient in some connected subgroup H* o/ H .

Proof. The first two paragraphs of L. AUSLANDER'S proof ([2], pp. 279-280) show that , after conjugat ion by an element of H , D* c W. T where W is a connected subgroup of H and T is a torus in C which centralizes W. For the sharpening, we replace the th i rd paragraph of a slight variant . D* is finitely generated because it is discrete in the connected solvable group W. T ([5], Th. 1'), so D*/[D*, D*] is a finitely generated abelian group. Thus D*/[D*, D*] --~ A ' • B I where A' is the torsion subgroup. [D*, D*] c W because T is abelian and centralizes W ; thus the project ion

f : D* ---> D*/[D*, D*] maps A ~-- D ~ T

isomorphical ly onto A ' ; it follows t h a t D ~ A • B where B-- - - f -~(B ' ) . Now let g : W. T --> W be the .projection and define H * to be the smallest analyt ic subgroup of W which contains g (B). g maps B isomorphically onto g(B) , g(B) is discrete in H * because T is compact , and it is s tandard tha t H*/g(B) is compact . Q.E.D.

3. 5. The deformation of the free abelian subgroup and the corresponding quot ient manifold is given by:

O n l o c a l l y s y m m e t r i c s p a c e s o f n o n - n e g a t i v e c u r v a t u r e 273

Deformat ion Theorem. Let G ~ S . C be a semidirect product o/ LxE groups, let D be a torsion/ree subgroup o/ G with generating set {dl, . . . , d~} such that, given d E D, there is a unique set (ui} o/integers such that d ~-- ~ldUldU2~2 " " " d u n *

suppose that D acts/reely and properly discontinuously on G/C by d : g C ~ d g C , and assume that the projection o / D into C lies in a torus A which centralizes D . Write d i ~- 8 i a i with s~ e S and a i E A , choose elements X~ in the LIE algebra o] A such that a i ~ exp(Xi ) , de/ine d~t) z d i . e x p ( - tXi) ~-- s i �9 exp ((1 -- t )Xi ) , and let D (~) be the group generated by {d~t)). Then

1. D (~ = D a n d D (1) c S . 2. I / K is a closed subgroup o/ C, so P ~- G /K is an analytic mani/old on

which G acts by g : h K ~ g h K , then each D (~) acts ]reely and properly discontinuously on P ; in particular, the projections P - + P / D (t~ are coverings o/ analytic mani/olds.

3. The maps d~ r) --> d! s) de/ine isomorphisms (the "de/ormation isomorphisms") o/ D (r) onto D (s) , and these isomorphism8 induce analytic homeomorphisms o/ P / D (r) onto P / D (~.

4. P / D is analytically homeomorphic to (S /D a)) • (C/K) .

Proo/. The first s t a t emen t is obvious. I f ~-d is a group relat ion and

U - d ( s ~ , . . . , s ~ ) : 1, then U--d(d~, . . . ,d~)~A

because A is abeliau, A contains the a~, and A centralizes the 8 i. But

D , - , A c D , - , C = - { 1 }

because D acts freely on G/C ; thus 'Jd(dl . . . . . dn) = 1 . This shows tha t the d~ sat isfy every relat ion satisfied b y the s~; it follows tha t d~t)-+ d~ induces a h o m o m o r p h i s m of D (~) onto D . E v e r y e lement of D (~) has some expression d (t) Ul d (t) u2 ( 1 ) ( 2 ) . . . (d~)) ~ because the s i sat isfy every relat ion satisfied b y the

�9 , �9 Un . d~, and every e lement of D has unique expression d~ l d ~ d~ , it follows tha t the ep imorphism D (~) -+ D is an isomorphism�9 This gives the defor- ma t ion isomorphisms.

For the second s t a t ement , we note t ha t D (~) c G acts freely and proper ly discont inuously on G/C, if and only if D (~) c S . A acts f reely and proper ly discont inuously on (S . A ) / A . As A is compact , and as D (t) is discrete (because D (1) is discrete in S, consequence of proper discont inui ty of D on G/C) and torsionfree (because it is i somorphic to the torsionfree group D), D (t~ mus t be free and proper ly discontinuous on (S . A ) / A . This proves the second s t a t ement ; the th i rd follows because the deformat ions are along analyt ic arcs.

For the last s t a tement , view P / D (~) as the double coset space D ( t ) ~ G / K . Writ ing ~_ for analyt ic homeomorph i sm, we then have

P / D ~ P / D (~) ~ (D(~)~S) �9 (C/K)

18 CM]~ vol. 37

274 JOSEPH A. WOLF

Now observe tha t ( s , c ) - + s c induces S x C ~ G - - ~ - S . C and s - + s -1 induces D~I)~S ~--- S/DII~; it follows t ha t P/D ~ (S/D ~1~) .'< (C/K). Q.E.D.

3 .6 . The finite covering. Iden t i fy ~ (N) with the group P of deck transfor- mations of the universal RI~.~AN~ian covering M - * N . M - - - - M o X M' where M 0 is a EucLIDean space and M' is a product of irreducible RIEMAN~ian symmetr ic spaces, for N is complete, connected and locally symmetr ic . As ~V has every sectional curva ture _>_ 0, the same is t rue for M ' ; it follows tha t M ' is compact because a noncompact irreducible RIEMANNian symmetr ic space has a negat ive sectional curvature . In part icular , the full group of isometrics I (M' ) is compact .

I(M0) is the ordinary EccmDean group on n----dim. M o variables, and m a y be viewed as a semidirect p roduc t R ~. 0 (n) where R ~ is the vector group and 0 (n) is the orthogonal group. This allows us to view I(M)---- = I(Mo) X I (M' ) as a semidireet p roduc t R ~. C where C ----- O(n) x I (M ' ) . A s / ' is a discrete subgroup of I (M), Proposi t ion 3.4 gives a finitely generated free abelian subgroup zJ of finite index i n /1 corresponding to the group B there.

B y construction, the project ion of A into C lies in a torus. The condition of Theorem 3.5 for expression of elements in terms of generators is obvious for finitely generated free abelian groups, zl acts freely and proper ly discon- t inuously on I (M) /C because C is compact and A is discrete and torsion free. Now Theorem 3.5 shows tha t M/A is real analyt ical ly homeomorphic to (Mo/A') X M' where/1 ' is a discrete group of pure translat ions of M 0 which is isomorphic to zJ. Define N ' = (Mo//l') X M ' and recall t ha t M/A -+ M / I ' = N is a finite RIEMA~ian covering. This proves the first s ta tement of Theorem 3.1.

3 .7 . The deformation retraction of N onto a compact submanifold is accom- plished by a deformat ion o f / ~ onto another group / " , followed b y a / " - e q u i - var ian t deformat ion re t ract ion of M. We retain the nota t ion F , M, M' , M 0 and A f rom w 3.6, except tha t we m a y replace A by the intersection of its conjugates in _P, and thus assume A normal in F .

E v e r y 9 , EF is of the form 9,0 • 9,' where 9,0 eI(Mo) and 9" EI (M' ) . For a choice of origin in M0, 9,0 is fur ther decomposed into (9,~, 9,,) where 9,~ e M 0 indicates a t ranslat ion and 9,, is a rotat ion. By construct ion of A, we may choose the origin so t ha t ~ : ~ - + ~ for every ~ r zJ. The origin so chosen, M o is identified with the vectorspaee R n, and we have an or thogonal direct sum decomposit ion M 0 ---- U + V where V is the subspace spanned b y the St. E v e r y 9,~ preserves V, and thus preserves U, because A is normal in F .

Given ? r we have 9 , t = - 9 , v + 9 , v with ? u r and 9 ,v~V. I f s is a real number , define 9,~'~ = (sg,v + 9,v, 9,,) X 9,' and l e t / ' , be the subgroup of I (M) generated by the 9,~'~. I t is easily checked t h a t 9'--> 9,~ defines an


isomorphism o f / 1 onto /1,. F, is discrete in I (M) because it contains A as a subgroup of finite index; thus / '~ acts properly discontinuously on M . Now if y(8) has a fixed point, i t must have finite order, whence y has finite order; it follows tha t ei ther y --~ 1 or y' has no fixed point; y(8) ~ 1 because the la t ter would prevent ~(8~ from having a fixed point. Thus F, acts freely on M . We now have a one parameter family of manifolds No : M/Fo which are analyt ical ly homeomorphic to N ~- N~. I t will be clear t ha t this isotopy of the metric of N is the ident i ty on a compact to ta l ly geodesic submanifold onto which No is retracted. For the proof of Theorem 3.1, then, we m a y replace N by N o. In other words, we may assume each y~ E V.

We h a v e M a s a RIEMAN~ianproduct U • V • M' where U a n d V a r e EccLIDean spaces with vectorspaee structure, and every y e F is of the form 71 • Y2 • ~/ where yl is a r o t a t i o n of U, y2 is a n i s o m e t r y of V, and F t is an i sometry of M ' . Define ], : M - ~ M by ]~(u, v, m') : (su, v, m'); /o is F-equivar iant because each y~ is a linear t ransformation. Thus ]~ induces a map g , : N - > N . This gives a deformation re t ract ion of N----g~(N) onto go(N). But go(N)~--/o(M)/F~-- (V • M')/F admits a covering by (V • • M')/A, and, as in w 3.7. , Theorem 3.5 shows tha t (V • M')/A is homeomorphic to (VIA') • M' where A' is the group of translat ions consisting of the ~ . V/A' is a torus, compact by definition of V; thus go(N) is compact .

We have now exhibited a deformat ion re t ract ion of N onto a compact submanifold. As singular homology and eohomology satisfy the h o m o to p y axiom, the bet t i numbers of N are finite, and the EULEn-PoINCAR]i characteristic ~ (N) is a well defined integer, in those theories.

Observe tha t the deformat ion o f / ' did not move any points of go (N). I t is now clear tha t go(N) is to ta l ly geodesic in N , for it is the image of a to ta l ly geodesic submanifold V • M ' of M.

3 .8 . Finiteness of the fundamental group. We have seen tha t the deformat ion retract ion go (N) admits a covering of some finite mult ipl ici ty r b y T • M', where T is a torus with ~I(T) isomorphic to the subgroup A of finite index in T'. As go (N), T and M r are compact manifolds, we have

z(N) = X(g0(N)) = r x ( T x M') = z (T )" z (M' ) .

Now suppose x(N) =~ 0. Then z (T) ~= 0 =/: g(M'). ~(T) =/= 0 means tha t

T is a single point, so x(N) = + z ( M ' ) and A = {1}. As A has finite index

in F, F = ~ ( N ) must be finite. Now z (M ' ) =/: 0 implies z (M ' ) > 0 because M ' is a quot ient space of a compact LIE group I (M' ) by a closed subgroup [8]; thus z(N) > O.

276 Jos~r~ A. WOLF

The second s t a t emen t , and the finiteness assert ion of the th i rd s ta tement , of Theo rem 3.1 are now proved.

Suppose again t h a t g (N) r 0. As F is finite, the ~0 of w 3.7 fo rm a finite group; it is classical t h a t some point of M 0 mus t be fixed under every Y0. Changing the origin in M 0, F is the group of isometries of M consisting of all / (a) • a , as a runs th rough a finite group 27 of isometries act ing freely on M ~ , where / is a h o m o m o r p h i s m y ' --> Y0 of X into the or thogonal group of M 0. Now z ( M ' ) r so g ( M ~ ) r where M ' - - - - M 1 • . . . • M~ is the decomposi t ion of M ~ into irreducible factors. I t follows t h a t Z (M~) ~ 0 [8] and eve ry group of isometries act ing freely on M I is finite Ell]. This proves the converse and finiteness condit ion of Theorem 3.2, and t ha t the manifold M/I" there is a complete connected locally symmet r i c RIEMAN~ian manifold wi th every sectional curva tu re ~ 0 and g ( M / F ) ~ O.

To complete the proofs of Theorems 3.1 and 3.2, now, we need only p rove t h a t every e lement of the group 27 of Theorem 3.2 has some order 2 ~+1 and induces a t r ans fo rmat ion of the t ype exhibi ted there.

3 .9 . I n order to s t udy the e lements of 27, we need some informat ion on fixed points :

Fixed Point Theorem. Let T be an isometry o/ a compact connected simply connected irreducible R I E M ~ i a n symmetric space S with Z(S):/: O. I / ~ has no/ixed point, then S is isometric to a real GI~ASSMA2ZN mani]old SO(4k)/SO(2k) • S0(2/c), k >: 2, and ~4 has a/ ixed point.

Proo/. Le t K be an i so t ropy subgroup of G --~ I (S). The ident i ty component K 0 contains a max i m a l torus of G o because Z (Go/Ko) = Z (S) :/: O, by SA- MELSON'S theorem [8], so eve ry e lement of Go is conjugate to an e lement of K 0. I n other words, eve ry e lement of G O has a fixed point. Le t t be the image of

in G/Go; ~ has a fixed poin t if t ~ : - 1. Suppose tha t ~2 has no fixed point . Then G/G o has an e lement of order

grea ter t h a n 2. I t follows f rom CARTA~'S construct ion of I (S) [4] t h a t

S : S0(4/~)/S0(2]r • S0(2k)

where /c ~ 2; if, fur ther , G/G o has an e lement u with u 4 :~ 1, then ]c : 2 a n d u h a s order 3. Bu t if t a : 1, t h e n v ~ e I o ( S ) : G 0 , so ~ is homotopic to the ident i ty . I t is known t h a t z mus t be fixed poin t in th is case ([14], w167 5 . 5 . 9 - - 5 .5 .10) , so ~ has a fixed point . This contradic ts t a : 1. The only o ther possibil i ty is t ha t t a : 1 and ~4 has a fixed point . Q.E.D.

3.10. 2-groups. We will see t h a t / ~ and 2: are 2-groups. I f 9 is an i somet ry of M ' : M~ • . . . • M t, then we have decomposi t ions


M ' = X 1 x . . . X X~, g = g l x . . . • g~

where gi is an i s o m e t r y o f Xi which cycl ical ly p e r m u t e s its i r reducible factors . Thus , u n d e r app ropr i a t e i sometr ic identif icat ions, we have X i ---- S~ • . . . X S~ (v~ factors) wi th Si irreducible, and

g~ : ( s l , . . . , s ~ ) - ~ (~iSv~, s~ . . . . . s~t_~)

gives the ac t ion o f g~ on Xi, for some i s o m e t r y z~ of S~. N o w if g has order m , t h e n each v~ m u s t divide m, say m = v~m~; g~ induces the t r a n s f o r m a t i o n T i • . . . • on X~, and z~i = 1.

Suppose now t h a t g has odd order m , a nd re ta in the n o t a t i o n above. E a c h 4 As z ( M ' ) # 0 , we h a v e zi m u s t have odd order, so z~ is a power o f zi.

X (S,) ~ 0, and The o re m 3 . 9 shows t h a t each vl has a f ixed po in t s i e S i. Define x~ = (s i, s i . . . . . si) �9 X~; then g i ( x i ) = X i. It follows that

x = (z~ , x , . . . . . z~) is a fixed po in t for g.

I f F is n o t a 2-group, t h e n it has an e lement ? of odd order m > 1. ? ~- ----/(a) • a where a has order m in Z . The cons idera t ions above show t h a t a has a fixed point , con t rad ic t ing the hypo thes i s t h a t 2: ac t f reely on M ' .

This p roves t h a t / ' a nd Z are 2-groups.

3 .11 . The fo rm of the group elements n o w comes easily. Le t 1 # g �9 2% T h e n g has some order m -~ 2 u+l u > 0. R e t a i n the n o t a t i o n o f w 3 .10 for the decompos i t ions o f M ' a n d g. Then m~ = 2 a~ and v~ = 2 b~ where ai -4- b, = u + 1. As g2~ has no fixed point , some g~ has no fixed poin t . F o r this index i , it is easi ly seen t h a t b,__< u , say u = b i + w, whence

has a fixed po in t on S~ if and on ly g ~ = z 2w • . . . • z~ ~. I t follows t h a t ~ i

i f k is a mul t ip le o f 2 w+~; b y T he o re m 3.9, w = 0 or l , and S~ is i sometr ic to S O ( 4 n ) / S O ( 2 n ) X S 0 ( 2 n ) (n ~ 2) in case w = 1.

This comple tes the p r o o f o f Theorems 3 .1 and 3.2. Q . E . D .

4. RIEMANNian nilmanifolds

and a s t ruc ture t heo rem for locally homogeneous spaces

The p r o o f of some par t s o f T h e o r e m 3.1 do no t m a k e full use of the h y p o - theses. W e will p rove the fol lowing ex tens ion to local ly homogeneous spaces.

4. 1. Theorem. Let M--> N be a universal R I E M , ~ i a n covering where M = M o • M ' , a nilpotent LzE group acts transitively by isometries on Mo,

and M ' is a compact RZEM,~sNian homogeneous mani/old. T h e n there is a real

analytic covering N'--> N o/ some finite mult ipl ici ty r > 0 where N ' : E • N " • M ' , N" is a compact coset space o/ a nilpotent LIE group by a

278 JOSEPH A. WOLF

discrete subgroup, and E is di/]eomorphic to a EvcLxDean space; i / M o is isometric to a E~cLzDean space, then N" is a to rns and there is a real analytic de/ormation retraction o/ N onto a compact totally geodesic submani/old which lilts to one o / the de/ormation retractions o/ N ' onto N" • M ' . I n particular, the E~LER-Poz~ca~ characteristic z ( N r) o / s ingu la r theory is a well de/ined

1 integer; now z*(N) --~ r z (N ' ) ' the so called rational EVLER-PoI~ca~ char-

acteristic o/ N , is a well de/ined non-negative integer, and y* (N) ~ y, (N) i/ M o is ElTCLiDean. I / Z * (Y) r O, then the ]undamental group ~1 (57) is/inite. I / Z * (N) :/= 0 and M r is RZEMx~ian symmetric, then 7fi (N) is a /inite 2-group.

I am indebted to DAVID B. A. EPSTEI~ for drawing my at tent ion to C.T.C. WALL's rational ECLER characteristic [ 10] and for suggesting a way of adapting it to this context. w 4.4 is based on conversations with him.

4. 2. RIEMANNian nilmanilolds are defined to be RIEMA~Nian manifolds which admit a transit ive nilpotent group of isometries. The structure of M 0 is clarified by:

Theorem. Let B be a positive de/inite bilinear /orm on the LIE algebra ~ o/ a connected nilpotent LIE group S , let K be the group o /a l l automorphisms of S which preserve B , and let X be S with the le/t invariant Rx~Ma~ian metric derived/rom B . Then X is a connected RZEMaNNian nilmani]old, I(X) is the semidirect product S . K acting by (s, k) : x -+ s �9 k (x ) , S is the nilradical (maximal connected normal nilpotent subgroup) o/ I o (X), S is a maximal connected nilpotent subgroup of Io(X), and S is the only transitive connected nilpotent subgroup o/ I (X) . Conversely, every connected R X E M ~ i a n nilmani/old is isometric to one o/ the mani/olds X described above.

Corollary. I / X is a connected RxE~aNNian nilmani/old and x E X , then the R i B M a ~ i a n structure on X de/ines a unique structure o/ nilpotent LIE group in which x -~ 1.

Corollary. Let ~ : Y - ~ X be a RIEMAN~ian covering where X i8 a RzE- M a ~ i a n nilmani/old, and let y E Y . Then Y is a R i ~ M a ~ i a n nilmani/old. Endow Y (resp. X ) with its canonical nilpotent L ~ group structure/or which y ~ 1 (resp. ~(y) ~ 1). Then ~ is an epimorphism o / L I ~ groups, and the deck trans/ormations of ~ are le/t translations by the elements o/ the kernel o/ ~.

Corollary. Let F be the group o/ deck trans/ormations o/ a universal R ~ - M ~ i a n covering X--> Y where X is a R~EMa~ian nilmani/old. Then these are equivalent:

1. Y is a R i ~ a ~ i a n nilmani/old. 2. Y is a R ~ M . a ~ i a n homogeneous mani/old.


3. F consists o/isometrie,~ o/constant displacement. 4. F consists o/isometrics o/ bounded displacement.

Corollary. Let zc : X .~ Z be a RzEM,i~ian covering where Z is compact and X is a R ~ E M ~ i a n nilmani/oId. Then z~ [actors into RZEMAN~ian coverings o~ : X--> Y and fl : Y--~ Z where Y is a compact nilmani/old and fl is o[ /inite multiplicity; Y is a RIEM.l~lvian nilmani/old i / and only i / i t is isometric to a ]lat torus.

We complete w 4.2 by deriving the Corollaries from the Theorem; the Theo- rem will be proved in w 4.3, and we will then go on to the proof of Theorem 4.1.

The first Corollary is clear because S c I (X) is unique and acts simply t ransi t ively on X in the Theorem. For the second, we give X its L~E s t ructure with g(y) ---- 1, let S c I (X) denote the left translations, and lift the action of S to Y after backing off to the universal covering group of S.

The third Corollary is a little more complicated. I t is clear t ha t (1) implies (2) and tha t (3) implies (4), and it is known [12] tha t (2) implies (3); thus we need only prove tha t (4) implies (1). Choose x E X and give X the ni lpotent LIE group s t ructure S in which x ~ 1. In the nota t ion of the Theorem, we must prove every element o f /~ to be central in S ; then S induces a transit ive nil- po ten t group of isometrics of Y, and (1) is proved.

Le t g e I (X) be an i sometry of bounded displacement, g = (s,/c) with s e S and k ~ K in the nota t ion of the Theorem. As K is compact, there is a compact set C c I ( X ) with h g h - l ~ C for every h ~ I ( X ) ; h = (t - i , 1) gives (t -~. s . k(t), k ) e C, and it follows tha t S has a compact set which contains t -~. k(t) for every t ~ S. The exponent ia l map exp: ~ - + S is a homeomorphism and ]c is linear on ~ ; it follows tha t k ~ 1 because the linear i sot ropy representat ion of K is faithful. Now g ~- (s, t ) . E v e r y (tst -~, 1) ~ C, so the closure of the conjugacy class o fs in S is compact . Let T be the centralizer of s i n S; now S / T i s compact . Let P � 9 with exp(P) - - - - s , and suppose Q �9 ~ ; it is easily seen tha t [P , Q] = 0 if and only if s commutes with exp (Q) ; thus T is connected. I t follows tha t S / T is homeomorphic to a EvcLH)ean space. As S / T is compact , we must have S ~ T ; thus s is central in S. This completes the proof of the third Corollary.

For the four th Corollary, let 27 be the group of deck t ransformations of the universal RIEMA~ian covering # : W--> Z and let A be the deck trans- formations of v : W - - > X . L.AusLiNDER has proved [1] tha t / ~ = E ~ Sw has finite index i n X ; as A c Z' and we have just seen A c Sw (for A is central in Sw), we have A c / ' . Now define Y ~- W/F, and the existence of a and fl is clear. I f Y is a R I E ~ N ~ i a n nilmanifold, then Y ~- Sw/F is a group, and so Sw/F is a torus. This proves the fourth Corollary.

280 Jos~P~ A. WOLF

4. 3. Proof of Theorem 4 .2 . Le t W be a connected RIEMA~Nian nilmanifold. There is a t rans i t ive n i lpotent group of isometries of W; its iden t i ty componen t T is t ransi t ive . T* will denote the closure of T in I (W) ; T* is nil- po ten t and t ransi t ive. Wri te W as a coset space T*/Z where Z is the i so t ropy subgroup of T * a t w e W. Z is compac t because T * is closed in I ( W ) ; thus Z is conta ined in a m a x i m a l compac t subgroup Z* of T*. Z* is connected because T* is connected, and a compac t connected subgroup of a nf lpotent LIE group can be seen to lie in the center b y looking a t the universal covering group and its exponent ia l mapping ; thus Z is central in T*. T* acts effectively on W; it follows tha t Z = (1} and T * is s imply t ransi t ive. As T c T * and T is t ransi t ive, this proves t h a t T is closed in I (W) and s imply t rans i t ive on W; it also proves t h a t T is m ax i m a l among the connected ni lpotent subgroups of I0 (W).

Suppose t h a t we can prove T to be contained in the nilradical N of Io(W ) . Then T -~ N , so T is normal in I ( W ) . I f H is the i so t ropy subgroup a t w e W, then H ~ T ~ {1) because T is s imply t ransi t ive, so I (W) is a semidirect p roduc t T �9 H . The representa t ion of H on the LIE algebra ~7 is equi- va len t to the linear i so t ropy represen ta t ion of H on the t angen tspace W~, and is thus faithful; now H m a y be viewed as a group of au tomorph i sms of T. I den t i fy T with W, viewing T as a LIE group with left invar ian t RIEMANNian metr ic specified b y some posi t ive definite bilinear fo rm A on ~:. Then H preserves A, and mus t contain every a u t o m o r p h i s m of T which preserves A because it contains every i somet ry of W which fixes w. Wri t ing I(W) ~- T . H , now, the act ion on T is necessari ly (t, h):v--> t. h(v). As the manifold X of Theorem 4 .2 is a R I E M a ~ i a n ni lmanifold under the group S there, this will p rove Theorem 4.2.

We now need only p rove T c N where N is the nilradical of I0(W ) . Le t ~ : W ' - + W be the universal RIEMAN~ian covering; we can lift the act ion of T on W to the act ion of a covering group T ' of T on W', and T ' will be t rans i t ive on W' . Le t F be the group of deck t rans format ions of the covering, let N ' be the nilradical of I0(W') , and let P be the normal izer o f / " in I ( W ' ) . ~ induces a h o m o m o r p h i s m :~* of P onto I ( W ) with kernel F, and T ' c P b y construction. I f T ' c N ' , t hen T ' c P ~ N ' , and the la t te r lies in the nilradieal N" of P . I t is clear t h a t ~*(N") ~-- N and ~ * ( T ' ) --~ T ; it will follow t h a t

T c N . N o w we assume W s imply connected, and need only prove T c N . Le t

R be the radical (maximal connected normal solvable subgroup) of Io(W ) . Then I0(W ) --~ S . R where S is a m ax i m a l connected semisimple subgroup. Let f l : Io(W)--> ad(S) be the composi t ion of tak ing quot ient b y R with the adjoint representa t ion of S/S ,-, R. E v e r y e lement g E I ( W ) has unique and

On locally symmetric spaces of non-negative curvature 28]

continuous decomposit ion g = th , t E T and h e H = isotropy at w; thus I0(W)/T is compact; it follows tha t a d ( S ) / f l ( T ) is compact. As f l (T ) is nil- po ten t and ad(S) is a product of centcrless simple LIE groups, ad(S) must be compact. This proves tha t S is compact.

The ident i ty component t t o is an isotropy subgroup and a maximal compact subgroup o f I 0(W); thus H 0 = S . H ' where H ' = ( H ~ R ) 0 is the iden t i ty component of the center of H 0 . Le t fl: I 0 (W)-->I0 ( W ) / N = U. N ~ H is a compact subgroup of N and is thus in a maximal compact subgroup of N ; this maximal one is central in N , thus unique, and thus central in Io(W); it follows tha t N ~ H = {1} so U = S R ' where R I = R / N . R' is abe l ianbecause [ R , R ] is ni lpotent and normal and thus in N ; it follows tha t R' = H ' • V where V is a vector group stable under S. Le t M = fl- l(V). M is a dosed normal subgroup of I 0 (W) such tha t I o (W)/M is compact and I 0 (W) is semidirect product M . H 0. Thus dim. M = d i m . Io(W ) - dim. H 0 = d i m . T. Let

: I0(W) - - > I o ( W ) / M .

T M is closed in Io(W) because T and M are closed and M is normal. Thus o~(T) = ( T M ) / M ---- T / T ~ M is a torus. On the other hand, T ~ M is con- nec ted because it is an analyt ic subgroup of T . As T is connected, simply connected and nilpotent, it follows tha t 0r = T / T , ~ M is homeomorphic to a EucLIDean space. Thus a (T) = {1}. This proves T c M. As they are connected groups of the same dimension, t hey must be equal. In particular, T is normal in I 0(W). This proves T c N , completing the proof of Theorem 4.2. Q . E . D .

Remark . Theorem 4.2 shows tha t the notion of RIEMAN~ian nilmanifo]d is bu t a mild generalization of the notion of RIEMA~'ian homogeneous manifold of constant curvature zero. The essential par t of the proof was exhibi t ing of V above. This was essentially done by reducing to the case of constant zero curvature .

Remark . One might define a t~IEMANNian solvmanifold to be a RI~MANNian manifold which admits a t ransi t ive solvable group of isometries, but the IWASAWA decomposition shows tha t this notion is not ve ry restrictive. For example, a RIEMA~Nian symmetr ic space with every sectional curvature =< 0 is a R I E M a ~ i a n solvmanifold.

4. 4. Rational E[JLEn-POINCAng characteristic. All spaces are connected, locally arcwise connected, locally simply connected, and with a basepoint which will generally not be mentioned. Le t C be the family of finite C W complexes, C the family of spaces homotopy equivalent (respecting basepoints)

282 Jos~r~ A. WonF

to an element of CO, and C ~ the family of spaces which admit a finite covering by an element of ~ . Given X e ~ , we have the EULER-POINCA~]~ characteristic (of singular theory) Z (X) = Z (Y) where X ~ Y e CO.

Proposition. I[ Z ~ C ~ so Z admits a covering o] some finite multiplicity

r :> 0 by some X ~ C t , then Z* (Z) = ~ z(X) is a well defined rational number,

which we will call the rational EvzER-Poz~vc~ characteristic o/ Z . I] Z 1 and Z~ ~ CO*, then z*(Z1 • Z2) = z*(Z1)z*(Z2). I[ Z 1 E CO* admits a t-[old covering by a space Z2, then Z2 E CO* and z*(Z2) =- tz*(Z1) .

The main step in the proof is:

Lemma. Given a finite covering g : (U, u) --> ( X , x) and a homotopy equivalence h : (X , x) --~ ( Y, y) of spaces with basepoint, let a : (V, v) --~ ( Y, y) be the covering with azx ( V, v) = hg~l (U, u). Then there is a homototry equivalence b : (U, u) --~ (V, v) which covers h.

To prove the Lemma, one defines b by b (u) --~ v and by defining b to cover h along any arc s tar t ing at u which is the lift of an arc start ing at x ; b is well defined because of the condition on fundamenta l groups. Let h' : ( Y, y) --> (X, x be a hom otopy inverse to h, and let b ' : (V,v)---> ( U , u ) be the map covering h ' , defined from h' as b was defined from h; it is easily seen tha t b' is a homotopy inverse to b.

Proo/o[ Proposition. To see tha t Z* (Z) is well defined, choose z eZ and r~-fold

coverings [i : (Xi, xi)-> (Z, z), X~ E C ~ ; we must prove 1 z ( X 1 ) = 1 r I r~ Z (X~).

St = [~nl(Xi, xi) is a subgroup of finite index r i in g l (Z, z); thus S = $1r 2 is a subgroup of some finite index sir ~ -= s~r 2 in g l (Z , z). This gives s~-fold coverings gi : (Ui, u~)--> (X~, Xl) with ]ig~gl(Ui, ui) = S . We have homo- t o p y equivalences h~ : (X~, xi) --> ( Yi, Y~) with Yi E Co; if a~ : ( V~, vi)->( Y~, y~) are the s~-fold coverings with a~u~(V~, v~) ---- h~g~z~(U~, u~), then it is obvious tha t V~ ~ C ~ and the L e m m a gives homotopy equivalences b~ : (U~, u~) ---> (V~, vi) �9 Thus U ~ e ~ ' and z ( U ~ ) = s ~ z ( X ~ ) . Now [ ig~: (U~,u i ) - -~ (Z , z ) are coverings with [~g~nx(U~, ui) = S; thus U~ is homeomorphic to U~; it i0110ws

tha t s~g(Xt) = s~g(X,) . Dividing by r181 - - - - r282, we have 1 1 r--l- Z ( /1 ) : r--2 Z (X$), and Z* (Z) is well defined. The other s ta tements follow easily f rom the corresponding s ta tements in CO, but we must use the Lem m a to prove Z, e CO* in the last s ta tement . Q . E . D .

4. 5. Proof of Theorem 4. 1. Le t F be the group of deck t ransformat ions of the universal R I ~ . ~ a ~ i a n covering M = M 0 • M ' -~ N of Theorem 4.1.


M o is a connected s imply connected RIEMANzr nilmaniibld; the same state- men t follows for each of its irreducible factors, so these irreducible factors are homeomorphic to EUCLIDean spaces by Theorem 4.2. As M ' is compact , none of its irreducible factors can be isometric to an irreducible fac tor of M o. Thus I ( M ) : I(M0) • I ( M ' ) . Theorem 4.2 shows tha t I(M0) is a semidirect product S . K where S is a connected s imply connected ni lpotent LIE group and K is compact . This allows us to view I ( M ) as a semidirect p roduc t S . C where C ~- K • I ( M ' ) is compact . Proposi t ion 3 .4 now provides a torsionfree subgroup A of finite index in F, an analyt ic subgroup S' c S, and a tora l subgroup T c C which centralizes S ' , such t h a t A c S ' . T and S ' /A ' is compac t where A' is the project ion of A on S ' .

We now need

Lemma . Let D be a discrete subgroup o/ a connected simply connected nilpotent LIE group U with U/D compact. Then D is torsion/roe and has a generating set ( a l l , . . . , dn} such that, given d ~ D, there is a unique set (v~} o/integers with d ---- "tvl "tv2 d~n

Proo /o /Lemma. D is torsionfree because U is torsionfree. Le t r be the length of the lower central series of U; let Z be the center of U. D ~ Z is the center of D because an au tomorph i sm of U is t r ivial if and only if it is t r ivial on D . As D is discrete, it follows tha t D Z is closed, so the i m a g e o f Z in U/D is closed, whence Z/(D ,~ Z) is compact . Le t { d l , . . . , d~} generate the free abelian

f group D ~ Z. B y induction on r , we have a generat ing set (d~+ 1 . . . . . dtn} of the requisi te sort for the group D/(D,-, Z) in U/Z. Let d~+ i be a n y

f element of D mapp ing onto d~+ i . Q . E . D . The L e m m a shows tha t A, being isomorphic to A' under the project ion of

S ' - T onto S ' , satisfies the conditions on generators of the discrete group of Theorem 3.5. The project ion of A on C lies in the torus T , and the act ion of A is free and proper ly discontinuous on (S . C)/C because A is discrete and torsionfree while C is compact . Thus M / A is analyt ical ly homeomorphic to M / A ' b y Theorem 3.5 .This provides the finite real analyt ic covering

N' -~ M / A ' - + M / F = N .

N ' = (S/A') • M ' , and S/A ' is homeomorphic to E x (S'/A') where E is homeomorphic to a EucLiDean space. Let N # ~- S ' /A ' , and the decomposition N - - - - E • N" • M ' is exhibited.

Le t r be the mult ipl ic i ty of the covering N ' -+ N . I f F is infinite, then A' 1 r,~

is nontr iv ia l and [6] z ( N ~) = 0. Thus z * ( N ) = x ( N ~) = r Z ( - h ) z (M ' ) = 0.

I f F is finite, t hen the project ion of F on I (M0) mus t have a s t a t ionary point

284 Jos~P~ A. WOLF

because the maximal compact subgroups of I(M0) are isotropy subgroups; thus /~ projects isomorphically onto a subgroup 27 of I (M' ) which acts freely on M ' . I f t is the common order of F and 27, then M' --> M'/27 is a covering

1 of mult ipl ici ty t. We have z(M'/27) I ~ - z ( M ' ) because M' is compact,

1 1 whence t divides z ( M ' ) . Now Z*(N} = -~ z (M) = ~- z (Mo)z (M' ) ~ -[ z(M') is an integer ~ 0.

We have proved tha t )~*(N) is an integer =~ 0 and tha t Z*(N) ~ 0 implies finiteness of :Y~l (N). If ~* (N) :~ 0 and M ' is RIF~MA~Nian symmetric , then ~I(N) is a finite 2-group as in w 3.10. Similarly, the ret ract ion of N when M o is E v c ~ D e a n is exhibited as in w 3. This completes the proof of Theorem 4.1. Q . E . D .

5. Classification in the irreducible case

We will classify (up to global isometry) the complete connected locally irreducible locally symmetr ic RIEMANNian manifolds of nonzero characterist ic and all curvatures > 0. This is the first step in implementing Theorems 3.1, 3 .2 and 4.1.

5 .1 . The candidates for consideration are not numerous :

Theorem. Let S be a compact connected simply connected irreducible RzE- MAN,Jan symmetric mani/old with Z (S) ~ 0, and suppose that S has a fixed point /tee isometry. Then S is a GtLISSMA2~ mani]old, SO(2n)/U(n) with n > 2 , Sp(n)/U(n) with n > 1, ET/AT, or ET/E 6. T 1.

Remark. Here A 7 is a subgroup SU(8)/{=]= I} in the compact simply connected exceptional group ET, and E 6. T 1 : (E 6 • T1)/{1, z, z 2} where T 1 is a circle group and z = (z', z"), each component of order 3 and z' central in E 6. GRAXSMAN~ mani/old means real, complex or quaternion GRASSMANN manifold, and we use oriented subspaces for real GnASSMA~ manifolds.

Proo]. Let K be an isotropy subgroup of G = I (S) . Bo th groups are compact , and rank. K = rank. G because g (S) =fi 0. In part icular, every element of G O has a fixed point on S. Thus we need only examine the cases where G =fi Go. According to CARTA]q [4], these are, besides the ones ment ioned in the s t a t emen t of the Theorem, only E~/{SU(6) x SU(2)/discrete} and E6/{S0(10 ) x S0(2)/discrete}. We will check that , for both of these spaces, eve ry i somet ry has a fixed point. Theorem 5.1 will then be proven.

5 .2 . Le t M be a symmetr ic space E6/{SU(6 ) • SU(2)/discrete} or EG/{S0 (10) • SO (2)/discrete}, and let K be an iso~ropy subgroup of G ~ I(M).


Then K --~ K o v a K o and G = G o " aGo where conjugat ion b y o~ induces outer au tomorph i sms bo th on K o and G o. For conjugat ion b y a is outer on Ko b y construct ion of I ( M ) [4]. Now let ~ : EG-+ G o be the project ion; the kernel D of ~ is the center of E6, cyclic of order 3, and ~-l(K0) is the centralizer of an e lement s ~ E 6 wi th s 2 ~ D . As D has odd order, we m a y assume s 2 = 1. I t follows tha t K 0 is its own normalizer in G 0. I f conjugat ion b y a were inner on Gu, it would be inner on Ko; this it not the case.

Now let A and B be the centralizers of a in G O and Ko, respectively. Checking bo th cases, we see t ha t both A and B have r a n k 4. I t follows tha t B contains a max ima l torus T of A .

Le t g ~ G. I f g ~ G o, then we know tha t g has a fixed point because r a n k . K = = r a n k . G. I f g ~ G O , then gEc~G o. Then, if V is a max ima l forus of A , hgh -1 ~o~V for some h ~ G o ([9], Th. on p. 57). Le t V be the max ima l torus T above. Then V c K 0, so hgh -1 e a K . This shows tha t g has a fixed point , p roving Theorem 5.1. Q . E . D .

5. 3. Space forms of GnnssMANN manifolds. Theorem 5.1 tells us which spaces should be studied in order to find the groups A of isometries act ing freely on a compac t irreducible s imply connected symmet r i c space S with g (S) =/= 0. Classification of these groups A up to conjugacy in I (S) is the same as classification of the space forms S/A of S up to isometry. I n [13] we solved the compl ica ted c a s e - t h e case where S is a GRASS~AN~ manifold. For the convenience of the reader, we will recall the results.

Le t F be a field R (real), C (complex) or I t (quaternion), and let F '~ denote a left posi t ive definite hermi t ian vectorspace of dimension n over F. I f O<fl<n, then the un i t a ry group U(n , F) of F ~ acts t rans i t ive ly on the set Gq, n(F) of q-dimensional subspaees (oriented if F --~ R) of P~. "~Ve exclude GI,~(R) and G2,a(R); then Gq,~(F) has a unique (up to a scalar multiple) U(n , F)- invar ian t RIEMA~)rian metric, and is a lways envisaged with t ha t metr ic; i t is s imply connected and RIE~AN~ian symmetr ic , and has topological dimension q(n -- q)r where r is the dimension o f F over R. The characteris t ic z(Gq,~(F)) =/= 0 except when F = R and q ( n - q) is odd.

Io(Gq,~(F)) is the group of mot ions induced b y U(n , F)0 (which is SO(n), U (n) or Sp (n)). I f q = n - q, we have an i somet ry fl g iven b y or thogonal complementa t ion (and consistent with or ienta t ion if F = R). In any ease, we use fl to assume q even i f q ( n - q) is even and F = R. I f F = C, we have an i somet ry a induced b y conjugat ion of C over R. I f F = R, we have an i somet ry co given by reversal of orientat ion. Let g~(0 ~ v g n) be the iso-

( /n O ) ~ U ( n , F ) I f n = 2m, let k b e t h e i s o m e t r y m e t r y induced b y - ~ - - I ~

286 JOSEPH A. WOLF (oI ) induced by --I ra 0 E U ( n , F ) . I f F = C , let h~(O~ v ~ n ) be the

"--(;/~_~ O ) ~ U ( n , ( ~ ) where a- - - -exp(~V ~ l / n ) isometry induced by . ~ a I , . Let

Z~ denote the cyclic group of order m. Now the space forms of GRASSMANN manifolds of nonzero characteristic are classified ([13], Theorems 1, 2, 3) by:

Theorem. Let A be a group o] isometrics acting ]reely on Gq,~(F), where q ( n - q) i s even ( sowemayapp ly f landassume q even) i] F ~- I t . I] A ~: (1}, then A is conjugate in I(Gq,,(F)) to one o/the groups:

F group isomorphic to conditions

H {1, figs} Z~ 2q ~ n, 0 ~ v < q

C {1, ~,k} Z2

(~ {1, flh~v_~} Z 2

It

It

R

{1, ~o}

{1, ~k} Z2

Z2

Z~

q and n ~ q odd

2q~-n, 0_~2v<q

2 q : n , l ~ 2 v - - l < q

n o n e

n even

2 q = n , O~2v<q

R {1, flg2v, co, oJ~g2v } Z2 • Z 2 2 q : n , O~2v<q

Each o/these groups acts ]reely on Gq.n (F), and any two distinct ones are not conjugate in I(Gq,n(F)).

5.4. The space forms of SO(4n)/U(2n) are given by:

Theorem. Let M be the R1~MA~ian symmetric mani/old SO(4n)/U(2n), n ~ 1, and let g~ and ~1 E Io(M ) be the respective isometrics induced by the

elements (I04"-~" 0 ) - and diag. {(9 01),..., (_o 01); (0-~)} o/ S0(4n). - - /2V

We have I(M) : Io(M) ~ v �9 Io(M) where ~ is central, T ~ ~- 1 and T ~ Io(M). Let A be a nontrivial group o/isometries acting/reely on M . Then A is conjugate in I(M) to one o] the n groups {1,vg~}, 0 _~ u < n, or to {1,ViOl}. Con- versely, these groups act ]reely on M and are mutually non-conjugate in I (M).

Proo]. Let G = I(M). We have a point p c M at which the symmetry is given by s = 4- diag. {(9 o 1) . . . . , ( 9 ~)}. Let K be the isotropy subgroup of G at p. Then Go= S O ( 4 n ) / { i I }, K o = U(2n)/{-V I }, G = G o ,-,~'Go

On locally symmetric spaces of non-nogative curvature 287

and K =- Ko ~ a �9 K 0 where conjugat ion of Go by ~ is the same as conjugat ion b y a--~ • - 1 ; . . . ; 1, - 1 } . Observe t ha t a c G o , define ~ = - ~ a , and note t ha t the first s t a t ement is proved.

Let b E G o . T h e n ~ h has a fixed point on M i f a n d only if u v h u -1-=o~k for some u EG o and / c e K o. As u v h u - ~ - - - - T u h u - l = - ~ a u h u -~, this is equivalent to a u h u - l = k, i.e., to h being Go-conjugate to an e lement of a K o. I f pr imes denote represent ing matrices, we observe t ha t a ' an t i commutes wi th s' and tha t U(2n) is the full centralizer of s ' in S0(4n) . Thus ~h has a fixed point if and only if some SO (4n)-conjugate of h' an t icommutes with s'.

Suppose fur ther t ha t h 2 ~-- 1. Then h' has square • I . Suppose first t h a t h '2 ---- I ; then h' is conjugate to some gv ~, and we m a y assume v _~ n because h r m a y be replaced b y its negat ive. I f ~h has a fixed point, then s' mus t ex- change the eigenspaces of -~1 and of - 1 for some conjugate of h', and it follows tha t v --~ 2n . On the other hand, if v = 2n , then h' is conjugate to a ' and it follows tha t Th has a fixed point .

Now suppose h '2 = -- I . Thus h' is S0(4n) -con juga te to k~ or to s'. Ob- serve t ha t k~ and s ' are not conjugate in SO(4n), even though they are conjugate in 0 (4n). I f vh has a fixed point, then we m a y conjugate and assume t h a t h' an t i commutes with s ' . Now s' and h' generate a quaternion algebra, and it is easily seen t ha t they are SO (4n)-conjugate. On the other hand, if h ~ is conjugate to s', then we m a y assume t h a t they generate a quaternion algebra; this done, they an t i commute and ~h has a fixed point . Thus vh is fixed point free if and only if h ~ is SO (4n)-conjugate to/c~.

A has a t most one e lement in each componen t of I ( M ) . As A :fi {1}, i t follows tha t A = (1,~h} where (~h) 2 - - ~ h v h ~ - ~ 2 h ~ - - h 2-= 1, h e G o . The Theorem now follows. Q . E . D .

5 .5 . The space forms of S 0 ( 4 n ~ 2) /U(2n -~ 1) are given by :

Theorem. Let M be the Rl~M,~Zc~ian symmetric mani/old

S0(4n ~- 2) /U(2n -{- 1), n :> 1 .

Then I ( M ) = O ( 4 n - F - 2 ) / ( : J : I } andwehave i some t r i e s h,=- 4- ( /4~+2-~- 1.0)

o/ M . Every nontrivial group o/isometries acting/reely on M is conjugate in I ( M ) to one o/ the n groups (1, h~,+l}, 0 <= u ~ n. Conversely, these groups act ]reely on M and are mutually non-conjugate in I ( M ) .

Proo/. M has a point p a t which the s y m m e t r y is given b y

8 = + ag. ((_0 (_0

let K be the i so t ropy subgroup of G ---- I (M) a t 1 o. Then G o ~-- SO ( 2 m ) / ( • I}

288 JOSEPH A. WOLF

and K o - - U ( m ) / ( q - I } where we define m - - - - 2 n - ? 1. G - G o ~ c ~ . G O and K -~ K o ~ ~ . K 0 where conjugat ion of G o by ~ is the same as conjugation by a = • d i a g . { l , - 1 ; . . . ; 1 , - 1}. As this conjugation is an outer automorphism of G O (because m is odd) we m a y ident i fy c~ with a , viewing G as O(2m) /{rk I} and K as { U ( m ) , - , a . U ( m ) } / { • This proves the first s ta tement .

Given g e I ( M ) , g' will denote one of the two matrices in 0 (2m) representing g. I f h, (v odd) has a fixed point on M , then h~ is conjugate in 0 (2m) to an element h~ -~ a'k' for some k' E U(m), whence s 'h~s ' - l=--h~ I. This shows tha t s' exchanges the eigenspaces of ~- 1 and of -- 1 for h~, proving tha t v = m. I t follows tha t the groups {1, h2~+1} (0 ~ u < n) act freely on M . As t he y are obviously mutua l ly nonconjugate, the converse of the second s t a t ement is proven.

Le t A be a nontr ivial group of isometrics acting freely on M. As every e lement of G O has a fixed point, A = {1,g} with det. g ' = -- 1. f = 1 implies g ' ~ i I , whence g,2 ~ - k I because det. g ' ~ - 1, so g is conjugate to some h~ (v odd). We m a y take v ~ m because h~ is conjugate to h2~_~, and then v < m because g is not conjugate to a. The second s ta tement follows. Q . E . D .

5 .6 . The space forms of Sp(n)/U(n) are given by :

Theorem. Let M be the Rx~M,~v~ian symmetric mani/old Sp (n)/U (n), n > 1,

let Sp(n) beviewedasthegroupo/all g~U(2n) suchthat g J ~ g = J = ~ I ~ '

and let g, d o ( M ) be the isometry induced by diag. ( I . . . . -- I~, I . . . . - I~}eSp (n). We have I (M) ~-- Io(M ) ~ v . I o ( M ) where 7: is central, ~ ~ I and ~ ~Io(M) . Let A be a nontrivial group o] isometrics acting ]reety on M. Then A is conjugate

in I (M) to one o/the groups {1, ~g~}, 0_~ v < ~-. Conversely, these

groups act ]reely on M and are mutually non-conjugate in I ( M ) .

Proo]. Let G ~ - I ( M ) . Then G o : S p ( n ) / { : k I } and

(o o ) ~ G o s = + - ~ - l I ~

is the s y m m e t r y at some p e M. Le t K be the i sot ropy subgroup of G at p .

Then K o = U ( n ) / { • } where U (n) consists of all (0 ~~ which b is

an n • n un i t a ry matr ix , K : K o ~ . K o and G ~ G o ~ . G o , where conjugat ion of G o by ~ is the same as conjugation b y • J . As • J ~ Go, the first s t a tement is p roved by set t ing ~ ~ ~ . ( • J ) .

On locally symmetr ic spaces of non-negative curva ture 289

Let b E G O . As in w fixed point on M if and only if h is G o- conjugate to an element of (4- J) �9 Ko. Suppose tha t h ~ ~ 1, and let primes denote representing matrices. I f h '2 = - I , then h' is Sp(n)-conjugate to J , whence ~h has a fixed point. Now suppose h ' ~ = I . Then h is conjugate

to some g~. I f ' " conjugate to J k ' , k ' = (b 0 / , g~ is \ - tb_l/ then I - - ~ ( J k ' ) ~ =

= ( - ~ - l "b 0 ) ( 0--b0-1 ) . This - b . t b - 1 shows t b = - - b , whence J k ' =- -- b

o ) last is conjugate by = b_ ~ EU(2n) to ; it follows tha t v - = ~ b y

n counting eigenvalues. On the other hand, if v ---- -~, then it is not difficult to

see, using the WEYL group, tha t h is G0-conjugate to ( 4 - J ) . k for every

k ~ K o s u c h t h a t k~=(b0 0 ) tb_l and t b = - b.

The Theorem now follows. Q . E . D .

5 .7 . The space forms of E 7 / ( A 7 o r Ee" T ~) can be described, as in w167 5.4 - -5 .6 , in terms of the elements of square 1 in the group ad(E,) = ET/C where C is the center of ET. These elements are known:

Lemma (]~.CARTA~r [3]). The group ad(ET) has elements 1 = sz7 , s ~ ,

sz6 x ~1 and 8Ds x ,4 i of square 1 where the centralizer o[ s R in ad(ET) is o[

C~RT~t~ classi]ication type H ; these ]our elements are mutually non-conjugate in ad (E:) and any element of square 1 in ad (ET) is conjugate to one of them.

Complement to Lemma. Let ~ : E T - + ad(ET) be the projection and let SRI E ~-I(SH). Recall that C = K e r . ~ = {1 z} cyclic order two. Then (s~7)~ = = x = 1 a n d = x = z .

Proo/. The Lemma is CARTA~'S classification of RIEMA~Cian symmetr ic spaces M with I0(M ) = ad(Ev).

Let Z be the identify component of the centralizer of s H in ad(ET), observe t ha t Z' ~- ~z-l(Z) is connected because Z contains a maximal torus; let S and S' be the respective centers of Z and Z ' , and note tha t ~ : S' -~ S is

! 2- - to - -1 sending z to 1 and s~ to s H .

I f H = - E 7 then S' has order two, so (s~) ~-~ 1. I f H --~ D 6 • A~, then the universal covering group of Z' is Spin(12) •

x SU(2). That group has center isomorphic to Z2 x Z2 x Z~, and S' is a quotient of its center. Thus (s~) ~ = 1.

In the other cases, we look at the linear isotropy representation on the RIEMA~cian symmetric space ad (Ev)/Z. As this space is irreducible, it follows

19 CMtt vol. 37

290 J o s ~ A. WOLF

t h a t S = (1, sR} if H = A 7 and S is a circle group if H = Ee x T 1 . Looking a t ~ : S ' -+ S, i t is easily seen t h a t (s~7)2 = z.

Le t H = Ee )< T 1 and let Z" be the group Ee • T 1. Z" has center S" isomorphic to Z3 X T1; we represent the e lements of S" b y pairs (u a, v) where u generates the center of E e and v is a urdmodular complex number .

We have coverings Z " ~ - ~ Z ' - ~ Z , and S = z~fl(S ~) is a circle group. Thus Ker . (~fl) = L 1 • L2 where L2 is a finite cyclic subgroup of T ~ and L 1 is cyclic order 3 wi th a genera tor (u, w) where w a = 1. Now Ker . fl = L~ • L 3 where L 3 has index 2 in L~ ; thus we m a y choose fl such t ha t Ker . fl = L 1 and L2 is genera ted b y (1, - 1). I t follows t h a t z = f l ( ( 1 , - 1)) and s B = ~ r f l

((1, l /~-- 1)). This shows s~ to be fl((1, : k b / - 1)) , whence (s~H) 2 = z. Q.E.D. We can now enumera t e the space forms of ET/A~ and of ET/E6" T ~ :

Theorem. Let M be one o] the R~MAzc~ian symmetric mani/olds E~/A7 or ET/E6" T 1. We have I ( M ) = I o(M) ,~ 3. I o(M) where �9 is central, T 2 = 1 and T ~ I 0 (M). Let /1 be a nontrivial group o/isometrics acting/reely on M . Then either /1 = (1, 3} or /1 is conjugate in I ( M ) to {1,TSD6• These two groups act ]reely on M and are not conjugate in I (M).

Proo/. The first s t a t e m e n t is known E4], ~ being central because E~ admi ts no outer au tomorph i sm. Le t K be an i so t ropy subgroup of G = I (M). Then G o = a d ( E T ) , K = K 0 ~ c ~ ' K 0 and G = G 0 ~ . G O where ~ 2 = 1 and conjugat ion b y ~ is the same as conjugat ion b y a E Go; ~ = ~ a . Altering a b y an e lement of K o if necessary, we m a y assume t h a t a is conjugate to s~7.

As before, let C = (1, z} be the kernel of the project ion ~ : E7--> ad(E7) and let pr imes denote represent ing elements in Ev. Let A be the centralizer of a ' in E~; A ~ SU(8)/ ( :k I} as seen in the p roof of the complement to the

L e m m a , z is represented b y • ~ 1 �9 I s , and a' is represented b y

-t- e x p ( 2 ~ V - 1 / 8 ) . I s .

Le t hEGo, h ' 2 = z . Replacing h b y a conjugate, h ' E A and h ' i s r e p r e -

s e n t e d b y ~ ( e x p ( 2 ~ V 0 - - 1/8)Ip 0 ) where p - ~ q = 8. e x p ( 2 n V ~ - - 1 5/s)Iq

T h a t ma t r i x mus t have d e t e r m i n a n t + 1 ; it follows t h a t p and q are even, p = 2u and q = 2v. Again replacing h b y a conjugate, h' e A is represented

b y : kd i ag . {el,,, ~6I~, el , , , eSI~} where e = e x p ( 2 n W ~ - 1 /8 ) . Le t s be the s y m m e t r y to M a t the point a t which K is i so t ropy subgroup of

G. Al though s commutes with a because it commutes wi th ~, s ' cannot commu te wi th a ' because conjugat ion by a induces an outer au tomorph i sm of K0. Thus the c o m m u t a t o r [s ' , a ' ] = z. I t follows t h a t s ' normahzes A and

On locally symmetric spaces of non-negativo curvature 29 ]

tha t conjugation of A by s' is an involutive outer automorphism. Thus we m a y assume [3] 8' gs '-1 ~ ~g-1 _~ ~ for every g ~ A, or tha t s 'gs I-1 ~ j g j - 1

( J = ( 0 Io')) f ~ g E A " I t f ~ 1 7 6 t h a t s ' h ' s ' - l = h ' z " N ~

h I : h" a r . As s' a ' s '-1 : a r z, h" must commute with s ' . This implies ~ (h") r Ko because ~-l(Ko) is connected and is the centralizer of s' in E~.

We have now proved, given h ~ G O with h ~2 : z, tha t u h u -1 -~ a k for some

k ~ K o, uEGo. This gives a u h u -1 -~ k , i.e., ~ a u h u -1 : ~ k , i.e., ~ u h u - l e ~ K o ,

i.e., ~h conjugate to an element o f ~ K 0. Thus vh has a fixed point on M. Now let h~G0, h ~ 1. We will see tha t ~h has no fixed point on M .

For if it had a fixed point, we would have u T h u -~ ~ ~]c with u E Go and /c e K0. Then h would be conjugate to aIc ~ a K o. Replacing h by tha t conjugate, h ~ ~- a ~/c' with ]c' in the centralizer K ' ~-- ~-~ (K0) of s' in E7. Now B ~ A ~ K ~ is both the centralizer of s r in A and the centralizer of a ~ in K ' . Eve ry element o f a r K ' is conjugate to an element o f a ' B. For K o ,~ a K o is the centralizer of s in ad (E:); if T is a maximal torus of the centralizer of a inKo, then a result of DESIEBENTKAT, ([9], Th. on p. 57) shows tha t every element of a K o is K0-conjugate to an element of a T ; thus every element of a ' K ~ is conjugate to an element of a' �9 ~ -~ (T ) c a ' B . :Now we conjugate h and assume h ~ ~ a ~ ]d where It' commutes with both s' and a ' . Thus we have /c' c A . Let double primes denote elements of SU(8) representing elements of A : SU(8)/{~: I } . h" --- a~k " is conjugate (by s ~') to h'rz " ; thus -- I : h "~, and it is conjugate in SU(8) to (h'~z~) 2~- hH2z ~'2~- ( - - I ) ( - - I ) ~ I . This

being impossible, ~h cannot have a fixed point. Our group A ~ (1, ~h} where 1 ~ (~h) ~ ~ ~2h2 ~ h 2. Thus, by the Lemma,

h i s conjugate to 1, sD~• S~v or SE~• But h '~--- 1, as we have just

seen, because ~h has no fixed point; the Complement to the L e m m a now shows h conjugate to 1 or s ~ • On the other hand, the Complement

and the preceding paragraph show tha t {1, ~) and (1, vs~e • ~l} act freely

o n ~ .

5.8 . Combining Theorems 5.1, 5.3, 5.4, 5.5, 5.6 and 5.7, one has a global classification for the space forms of compact connected simply connected irreducible RIEMANNian symmetr ic manifolds of nonzero characteristic.

6. Reducibility and eommutativity

6. 1. Order. Let N be an irreducible compact connected simply connected RI]~MAN~ian symmetr ic manifold of nonzero characteristic. We have just seen tha t a group of isometries acting freely on N must be of order 1 , 2 or 4. We

292 Jos~.PE A. WoT.~

now define the order o/ N, written order. N, to be the maximal of the orders of the groups of isometries acting freely on N. This concept is useful for:

6.2. Commutativity Theorem. Let M r be a compact connected simply connected R I E M ~ i a n symmetric mani/old o/ nonzero characteristic. Then these are equivalent:

1. A group o/isometries acting/reely on M r is necessarily abelian. 2. A n y group o/ isometries acting /reely on M r is a direct product o/ some

number m ~ 0 o/groups Z~, or is cyclic o/ order 4. 3. I / o n e o/ the irreducible/actors of M r has order 4, then all the others have

order 1. I / M r has two isometric irreducible/actors o/ order 2, then all the others have order 1.

Complement to the Commutativity Theorem. Let N be an irreducible com- Tact connected R I E M ~ i a n symmetric mani/old o/ nonzero characteristic.

1. These are equivalent: (a) N has order 2. (b) Z2 acts/reely by isometries on N , but Za does not. (c) Z~ acts/reely by isometries on N , but i s • Z~ does not. (d) N is isometric to Gq,,(R) where n :fi 2q and q(n -- q) is even, or to

Gq,,(C) whereeither 2q = n or q(n - q ) isodd, orto G~,~(H), otto SO(2n)/U(n) where n :> 2, or to Sp(n)/U(n) where n>= 1, or to E7/AT, or to E7/E6" T 1.

2. These are equivalent: (a) N has order 4. (b) Z 4 acts/reely by isometries on N . (c) Z2 • Z~ acts/reely by isometries on N . (d) N is isometric to G,~,4n(R) where n > 1. Here Z~ denotes the cyclic group of order m. The Complement follows trivially from the results of w 5. The remainder

of w 6 is devoted to the proof of the Commutativity Theorem. As (2) obviously implies (1) there, we need only prove that (3) implies (2) and that (1) implies (3).

6. 3. The proof that (3) implies (2) is based on Theorem 3.2 and on

Lemma. Let A be a nontrivial group o/ isometries acting/reely on N • N where N is a complete connected s imply connected irreducible RZBM,t~Nian symmetric mani/old o/nonzero characteristic and order 2. Then A is isomorphic to Z~, Z2 • Z~ or Z4.

Proo/. A , ~ { I ( N ) • has order ~ 4 and has index ~_2 in A, by Theorem 3.2; it suffices to prove that A is not a nonabelian group of order 8.

Suppose A nonabelian of order 8. Then A is generated by an element ? of order 4 and an element ~ of order 2 or 4, where ~ ~ ~-1 _~ ~-~. By Theorem 3.2,


we m a y assume r to be given by (x, y) -~ (vy, x) where T is a fixed point free i sometry of order 2 on N . I f ~2 = 1, then ~(x, y) = (~lx, ~ y ) where ~ is an isometry of square 1 on N . Then ~y~ =- 7 -1 implies ~1 = �9 5~; it follows tha t 7 ~ ( x , y ) = (~lY, ~lx); thus (x, ~lx) is a fixed point for ?~ . This proves ~ ~ 1.

Now ~ must have order 4, and is thus given b y (x, y) -~ (~lY, ~2 x) where ~i are isometries of N . Thus a = (~ is given by (x, y)--~ (alx, a~y) where a, are isometries of N . By Theorem 3.2 we have a S ~-- 1. But a * =~ 1 because A is the quaternion group. The Lemma follows. Q.E.D.

We will prove tha t (3) implies (2) in Theorem 6.2. Assume (3) and let P be a group of isometries acting freely on M ' . I f ~ , r then ~4_~ 1 b y Theorem 3.2. I f F has no element of order 4, it must be a product of groups Z2, and we are done. Now suppose tha t P has an element of order 4. By our assumption (3) and by Theorem 3.2, there is a RIv.MAN~ian product decomposition M' ~-- S • X where X is a product of irreducible manifolds of order 1 and either S is irreducible with order. S =- 4 or S =- S 1 • $2, $I isometric to S~, with order. S~ ~-- 2. Le t A be the restriction o f / ' to S. The restriction F - + A is an isomorphism; thus it suffices to prove A isomorphic to Z4.

Observe tha t A acts freely on S. I f S is irreducible of order 4, then A ___~ Z4 by the Complement, by Theorem 5.3, and because it contains an element of order 4. I f S is reducible, then A ~-- Z4 by the Lemma above.

6 .4 . To prove tha t (1) implies (3) it suffices to exhibit a noncommuta t ive group ofisometr ies acting freely on a direct factor of M r , in case the conditions of (3) do no t hold. For this noncommuta t ive group will then act freely b y isometrics on M ' . Thus we need only take compact connected simply connected irreducible RIEMAZ~z~ian symmetr ic manifolds N and L , order. L > 1, and prove :

I / order. N = 4, then there is a noncomrautative group o/ isometries acting /reely on N • L . I / order. N -= 2, then there i8 a noncommutative group o/ isometries acting/reely on N • N • L .

We will construct examples of such groups which are dihedral groups of order 8.

Suppose tha t N has order 4. Then N = (]2~,4~(R), n ~ 2, and (Theorem 5.3) flg**-i = ~ generates a cyclic group of order 4 of isometries acting freely on N . Let ~ = v • I r • L). Choose a fixed point free i s o m e t r y ~ of order 2 on L and define ~ ---- g2,-1 • 7. Then ~ has order 4, ~ has order 2, and (~7~ ~-- 7 - 1 because g2~-~flg2~,-~ = o~fl. :Now

r = 7 , r

is the (dihedral) group generated by ? and 6. The powers of ~ act freely on the

294 JOSEF~ A. WOLF

N-componen t , and the last four e lements move every L-coordinate . Thus F is a n o n c o m m u t a t i v e group of isometries act ing freely on N • L .

Suppose t h a t N has order 2. Le t v and ~ be involut ive fixed point free isometr ies of N and L , respectively. W e define e lements ? and ~ o f I ( N • N • L) b y ? ( x , y , z ) = - (vy, x , z ) and ~(x, y , z) ---- (vx, y , ~ z ) . ? has order 4 and its powers act freely. ~ has order 2 and any ~?~ moves the L-coordinate . 5y5 = ?-1 is easily checked. Thus the group F genera ted b y ? and 5 is a n o n c o m m u t a t i v e group act ing freely b y isometries on N • N • L .

Theorem 6.2 is now proven.

Remark. The other n o n c o m m u t a t i v e group of order 8, the quaternion group, can act freely b y isometrics on N • N • N X N where N is as above with order. N > 1.

6. 5. Corollary. Let M - + N be the universal RtEMaN2vian covering o/ a complete connected locally symmetric Rz~MaN~ian manifold N with every sectional curvature >= 0 and characteristic Z (N) ~ O. Suppose, i/ one o/ the compact irreducible ]actors o/ M has order 4, that all the others have order 1; suppose, i/ M has a pair o/ isometric compact irreducible /actors o/ order 2, that all the others have order 1. Then the/undamental group ~1 (N) is a/ ini te direct product o/groups Z2, or is cyclic o/order 4.

This follows immedia t e ly f rom Theorems 3.2 and 6.2. We can give a good descript ion of the manifold N of the Corollary. One has

M - ~ Mo X M1 X . . . X M t

where M 0 is a E~cLIDean space 1% ~ and each M i (i > 0) is compac t and irreducible wi th Z (M,) > 0. I f ~ (N) --~- Z 4, there are two sorts of possibilities: some M, has order 4 or two isometr ic M , have order 2. We p e r m u t e the M~ and obta in M -~ M 0 • S • X where S is irreducible of order 4 or the product of two isometric irreducible manifolds S i of order 2. N =- M / F where F i s genera ted b y an e lement ? = ? 0 • ?s • ? x , ?04= 1, ?~----- 1, and ?s is g i v e n b y : I f S i s irreducible, S-----G2~,4~(R) with n=> 2, then ?s is conjugate to an i somet ry flg2,-i of S. I n the other case, ? s is conjugate to an iso- m e r r y (sl, s2) -~ (Ts2, Sl) of S = $1 • $2 where T is a fixed point free invo- lu t ive i somet ry of S 1 = S~.

Suppose Y/:I (N) r 24; t hen n 1 (N) is a p roduc t of some n u m b e r k ~ 0 of groups Zz, and N = M / F for a g roup F isomorphic to ~1 (N) and given as follows. F has generators {?x, . - . , ?~}. Suppose first t h a t some M~ (say S) is of order 4, or t h a t two isometric M i (say S 1 and S~; let S ---- S 1 • $2) are of order two. Then M = M0 • S • X , where X is a p roduc t of irreducible manifolds of order 1, k < 2 , and each Y ~ : Y ~ , o X y i . s X ?~.r ; Y~'--~Y~,s


is an isomorphism of F onto a group 27 of isometries acting freely on S and each ~'i,s preserves each S if S~ is a product , so the possibilites for 27 are given in w 5; the Y~,0 commute and have square 1, as do the F~,x. We now con- sider the other p o s s i b i l i t y - t h e case where no M~ has order 4 and no two M i of order 2 are isometric. Re-ordering the Mi, we may assume tha t M1, . . . , M k each has order 2 and is preserved by each ~;., and tha t Fj induces a fixed point free involutive isometry of M; . Then M ~ M 0 • M 1 • . . . • M k • X , ~j ~-- ?i,o • . . . • ~j,~ • V j , x , ~ - - > Yi,~ (e ~ O , 1 . . . . , / c , X ) is a homomorphism of F, and ~I,i has no fixed point.

T h e I n s t i t u t e / o r A d v a n c e d S t u d y

REFERENCES

[1] L. AUSLAI~DER, BIEB:ERB&CH'S theorem on space groups and discrete uni]orm subgroups o] LI~ groups, Annals of Math. 71 (1960), 579-590.

[2] L. AUSLAI~DER, BIEBERBACH'8 theorem on space groups and discrete uni]orm subgroups o] LIE groups, I I , American J.Math. 83 (1961), 276-280.

[3] ~. C~TA~, Sur une classe remarquable d'espacea de RIE~A~N, Bull. Soc. Math. de France 54 (1926), 214-264; 55 (1927), 114-134. Also Oeuvres Complbtes, part. 1, vol. 2, 587-660.

[4] t~. C~a~T~, Sur certaines /ormea RiE~aANNiennes remarquables des g~om~tries 5 groupe /onda- mental simple, Ann. Sci. Ec. Norm. Super. 44 (1927), 345-467. Also, Oeuvres Completes, part. 1, vol. 2, 867-990.

[5] G.D.MosTow, On the/undamental group o /a homogeneous space, Annals of Math. 66 (1957), 249-255.

[6] K. NOMIZU, On the cohomology o/compact homogeneous spaces o] nilpotent LIE groups, Annals of Math. 59 (1954), 531-538.

[7] G. DE R m ~ , Sur la r$ductibilit$ d'un eapace de RIEM~N, Comm.Math.Helv. 26 (1952), 328-344.

[8] H. S~ELSO~, On curvature and characteristic o/ homogeneous spaces, Michigan Math. J. 5 (1958), 13-18.

[9] J. DE SIE~E~THAT., Sur lea groupes de LIE non connexes. Comm.Math.Helv. 31 (1956), 41-89. [10] C . T . C . W ~ , Rational EULER characteristics, Proc. Cambridge Phil. Soe. 57 (1961), 162-184. [ 11] J .A. WOLF, The mani]olds covered by a Ri~MxN~ian homogeneous mani/old, American J. Math.

82 (1960), 661-688. [12] J.A.WoT,F, Sur la classification des vari~t~s RxE~Y, Niennea homog~nea ~ courbure constante,

C.r.Aead.Sci. Paris 250 (1960), 3443-3445. [13] J.A.WoL~', Space ]orms o] G R A S S ~ mani]olds, Canadian J .Math. 15 (1963), 193-205. [14] J.A.WoLF, Locally symmetric homogeneous spaces, Comm. Math. Helv. 37 (1962), 65-101.

(Received May 24, 1962)

on locally symmetric spaces of non-negative curvature and certain other locally homogeneous spaces

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