The Clifford-Klein Space Forms of Indefinite Metric Author(S): Joseph A

Annals of Mathematics

The Clifford-Klein Space Forms of Indefinite Metric Author(s): Joseph A. Wolf Reviewed work(s): Source: The Annals of Mathematics, Second Series, Vol. 75, No. 1 (Jan., 1962), pp. 77-80 Published by: Annals of Mathematics Stable URL: http://www.jstor.org/stable/1970420 . Accessed: 03/05/2012 14:54

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http://www.jstor.org ANNAL!S OF MIATHEMATICS Vol. 75, No. 1, January, 1962 Printed in Japan

THE CLIFFORD-KLEIN SPACE FORMS OF INDEFINITE METRIC

BY JOSEPH A. WOLF* (Received February 24, 1961) 1. Introduction The sphericalspace formproblem of Clifford-Kleinis the classification problemfor complete connected riemannian manifolds of constantpositive curvature. E. Calabi and L. Markushave recentlyconsidered the classificationproblem for completeconnected Lorenz n-manifoldsM," of constant positive curvature,and have [1; Theorems2 and 3] reducedit to the sphericalspace formproblem for riemannian (n - 1)-manifolds,when n ? 3. We will extendtheir ideas to moregeneral signaturesof metric, and reduce the classificationproblem for complete connectedpseudo- riemanniann-manifolds M,, of constantpositive curvature, and with s # n - 1 and 2s < n, to the sphericalspace formproblem for riemannian (n - s)-manifolds. As the spherical space form problemis solved in dimension3 [3], and is trivialin even dimensions,this gives a classification (up to global isometry)of the completeconnected pseudo-riemannian manifoldsM8' of constant positive curvature with s # n - 1, 2s < n, and eithern - s = 3 or n - s even. I am indebtedto L. Markus and E. Calabi for showingme the manu- scriptof theirpaper [1], whichis the basis forthis note.

2. Pseudo-riemannianmanifolds In orderto establishterminology and notation,we recall some basic facts about pseudo-riemannianmanifolds. A pseudo-riemannianmetric Q on a differentiablemanifold Mis a differentiablefield of non-degenerate bilinearforms Qp on the tangent-spacesMp of M; a pseudo-riemannian manifold is a differentiablemanifold with a pseudo-riemannianmetric. We will consideronly those pseudo-riemannianmanifolds (M, Q) where each Qp has the same signature', say -dxi - * - dx' + dx'l1 + ** * + dx2; (M, Q) is thendenoted MS". In the riemanniancase (s = 0), we will simply writeMn. A pseudo-riemannianmanifold Msn carries a uniquelinear con- nection-the Levi-Civitaconnection on the tangent-bundleof M8n-with torsionzero, and such that paralleltranslation is an isometryof tangent- spaces. In localcoordinates x-(x1, ... , xn),we set qij(x) = Qx((&I8xi),(0/1xj))

* The author is grateful for National Science Foundation Fellowship support while writ- ing this paper. This is automatic if M is connected. 77 78 JOSEPH A. WOLF and lowerindices of the curvaturetensor Ri k,(X)of the Levi-Civitacon- nection, using the classical formulaRijkl(X) = :qjm(x)Ri kt(x); M7 has constant curvature K if

RijkZ(X) = K[qik(x)qi,(x) - qik(x)qj,(x)] identicallyin x, forevery (i, j, k, 1)and everylocal coordinatesystem. M." is completeif its Levi-Civitaconnection is complete,i.e., if everygeodesic can be extendedto arbitraryvalues of the affineparameter. An isometry is a diffeomorphismd: N,4-+Mf"where everytangent map is a linearisom- etry; a pseudo-riemanniancovering is a coveringf: Nn -+ M;' of connectedpseudo-riemannian manifolds where every tangent map is a linear isometry. The indefiniteorthogonal group Os(n + 1) is the group of all linear transformationsof the real vectorspace R"+' whichpreserve the bilinear formbn(x, y) = -x-y-* * *-x~y. + x,+ly+y + * * * + x,+ly?1+y.The quadric bl(x, x) = 1 carries a pseudo-riemannianmetric such that it is a pseudo- riemannianmanifold Sn of constantcurvature +1, and Os(n + 1) is its full groupof isometries[4,? 4]. We defineSs to be Ss if s < n - 1, Sn-_ to be the universalpseudo-riemannian covering manifold of SnU1,and Sn to be the componentof (0, **, 0, 1) in Sn. We have proved[4, Theorem 5] that every complete connectedpseudo-riemannian manifold Mn of constantcurvature +1 admits a universal pseudo-riemanniancovering f: Sn -* M, and is thus isometric to the quotient manifold SSID where the subgroupD of the groupof isometriesof S9nis just the groupof deck transformationsof f. Thus we define a pseudo-spherical space form of signature(s, n - s) to be a completeconnected pseudo-riemannian mani- foldMst of constantcurvature + 1. A sphericalspace formis just a pseudo- sphericalspace formM" = Mn, and, in the terminologyof [1], a relativistic sphericalspace formis a pseudo-sphericalspace formMl.

3. The reduction This reductionwas inspiredby that of Calabi and Markus [1] where they do the case s = 1; the proofof our Theorem1 beingvirtually identi- cal to theirproof for s = 1, whilethe methodsof our Theorem2 are dif- ferentfrom their quite geometrictechniques.

THEOREM1. Let M: be a pseudo-spherical space form with s # n - 1 and 2s ? n. Then Mn has finite fundamental group. PROOF. Sn = 9Snbecause s < n - 1, so we have a universal pseudo- riemanniancovering f: Sn -* M84. The groupD of deck transformations of the coveringis isomorphicto the fundamentalgroup of Mn, so we must CLIFFORD-KLEIN SPACE FORMS 79 prove that D is finite. S: is given by -EIxI + EI+IX2 =1 where the x; are coordinates in R71+relative to a basis {v1, ***, v,,1}. Let V be the subspace spanned by {vats, *., v,?1}. If g is any non-singularlinear transformationof Rn+1' then 2s < n < n + 1 gives us dim(Vn gV) ?2 dim V- (n + 1) > 0 . If Sn5 denotes the "equatorial sphere" xi = x. =s = 0, Es+lxj = 1 in S8", and if g e D, it follows that g(Sn-s) meets Sn-s The rest of the argument follows [1]. If D were infinite,compactness of Sn-swould give a point p e 5n-S, a sequence {gij of distinct elements of D, and a sequence {pi} in Sn-s such that each gi(pi)e Sn-s and {gi(pi)} -*p We may pass to a subsequence and assume {pi} __p' e Sn-S. This gives an element g e D with g(p') = p; so {g-1gi(pJ} -* g-1(p) = p'. As all the gi are distinct, we may assume that none of the g-1gi is 1 e D. Thus every neighborhood of p' in Ssnmeets one of its transformsby an element #1 of D. This is impossible because D, being a group of deck transformations, acts freely and properlydiscontinuously on Ss. q.e.d. Notice that the product 0(s) x 0(n - s + 1) is a maximal compact subgroup of Os(n + 1), where 0(s) is the ordinary orthogonal group on the first s coordinates in Rn+1and 0(n - s + 1) is the ordinary orthogonal group on the last n - s + 1 coordinates in R LEMMA (E. Cartan). Every compact subgroup of Os(n + 1) is conjugate to a subgroup of 0(s) x 0(n - s + 1). PROOF. This follows from a technique of E. Cartan [2, ? 16]. Also, A. Borel has remarked that the lemma is just simultaneous diagonaliza- tion of two quadratic forms. q.e.d. THEOREM 2. Let Sn-sIG be a spherical space form, view G as the image of a faithful orthogonal representation f: H-* 0(n - s + 1) of an abstract finite group H, let g: H-* 0(s) be any orthogonal representation of H. and set D = (g + f)(H) c0 (s) x 0(n-s + 1) c Os(n + 1). Then D is free and properly discontinuous on Sn, SSnJDis a pseudo-spherical space form, and S,,/D has finite fundamental group if s # n -1. Conversely, let Mn be a pseudo-spherical space form with s # n - 1 and with finite fundamental group (automatic if 2s ? n). Then Mn is isometric to one of the manifolds SS'7D described above. PROOF. D is finite, thus properly discontinuous on Sn. If h e H and (g + f )(h) has an eigenvalue + 1, then either h = 1 or the corresponding eigenvector lies in the span of the first s coordinates on Rn l. Thus D acts freely on SSn. This shows that S,,/D is a pseudo-spherical space form. 80 JOSEPH A. WOLF

If s * n - 1, then S." is simplyconnected, and the fundamentalgroup of S.:/D is finite. For the secondpart of the theorem,let f: S." -- M." be the universal pseudo-riemanniancovering, and let D be the groupof deck transformations. As s * n - 1, D is a subgroupof 08(n + 1). D was assumedfinite, thus compact,although this is automatic if 2s < n. Using the lemma, we may replace D by a conjugate,thereby changing M," by an isometry, and assume D c 0(s) x 0(n - s + 1). Now writeevery a e D in the form (al, a2) witha, e 0(s) and a2 G0(n - s + 1). The {a2} generatea subgroup G of 0(n - s + 1) whichacts freelyon the equatorialsphere So-* in S."; thus S''-/G is a sphericalspace form. Set H = D, f(a)= a and g(a) = a,, and the theoremis proved. q.e.d. Theorems1 and 2 reduce the classificationproblem for completeconnected pseudo-riemannianmanifolds M." (s # n - 1 and 2s < n) of constant positivecurvature to the classificationproblem for completecon- nectedriemannian manifolds M,,-' of constantpositive curvature.

INSTITUTE FOR ADVANCED STUDY

REFERENCES

1. E. CALABi and L. MARKUS, Relativistic space forms, Ann. of Math., 75 (1962), 63-76. 2. :. CARTAN, Groupessimples coe et ouvertset la giometrieriemoannienne, J. Math. Pures Appl., 8 (1929), 1-33; OEuvres completes, part. 1, vol. 2, 1011-1044. 3. W. THRELFALL and H. SEFERT, Topologische Untersuchung der Diskontinuititsbereiche endlicher Bewegungsgruppen des dreidimensionalen sphiriachen Baume", Math. Ann., 104 (1930), 1-70; ibid, 107 (1932), 543-586. 4. J. A. WOLF, Homogeneous manifolds of constant curvature, Comment. Math. Helv., to appear.