Mathematische Zeitschrift 9 Springer-Verlag 1995

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1 Math. Z. 218,597~502 (1995) Mathematische Zeitschrift 9 Springer-Verlag 1995 On radius, systole, and positive Ricci curvature Frederick Wilhelm * Department of Mathematics, S.U.N.Y. at Stony Brook, Stony Brook, NY 11794, USA ( wilhelmmath.sunysb.edu) Received 28 June 1993; in final form 18 October 1993 A closed Riemannian n-manifold M with Ricci M > n - 1 has diameter < 7r ([M]), and equality holds only if M is isometric to the unit sphere S ~ ([Cn]). Given these results, it is natural to ask whether M is diffeomorphic to S n if the diameter of M is almost 7r. This question was answered negatively by Anderson and Otsu, who showed that even the topology of such a space can be different from the topology of the sphere ([A], [O1]). Thus one must have stronger hypotheses to prove a differentiable sphere theorem. There are already results along these lines in [O2], [PSZ], [Wm], [Wu], and [Y1]. We will prove generalizations of all of these theorems and provide a corresponding characterization of real projective space. Recall that the radius of a compact metric space, (X, dist) is defined by (Cf [GP2], [SY].) Radius X = min max dist(x, y). xex yex Theorem 1 (Radius Pinching Theorem) Let n C N and k c R. There is an c > 0 so that ifm is a closed Riemannian n-manifold with Ricci M > n- 1 sec M > k, and Radius M > 7r - ~, then M is diffeomorphic to S n. Moreover, the Lipshitz distance between M and S '~, distl(m,sn), converges to 0 as ~ ~ O. This generalizes Theorem 3 in [O2], Corollary 2 in [PSZ], and Theorems 9 and 28 in [Wm]. * Supported by a National Science Foundation Postdoctoral Fellowship

2 598 F. Wilhelm Recall that the first systole of a closed Riemannian manifold, Sys~ M, is the length of the shortest closed, noncontractible curve. As pointed out in [Wm], a lower bound for Sysl M implies a lower bound for the radius of the universal cover of M. This is the main idea in the proof of the following systole pinching theorem. Corollary 2 (Systole Pinching Theorem) Let n E N and k E. e > 0 so that ifm is a closed Riemannian n-manifold with There is an Ricci M >>_n-l, secm >k, and c~ > Sysl M > Tr - e, then M is diffeomorphic to RP n. Moreover, lim~0 distl(m, RP n) = O. This generalizes Theorem 10 in [Wm]. Before stating the next Theorem, we review the various notions of excess. Definition 3 Given two points p and q in a compact metric space X, the excess function for p and q, ep,q : X ----* R, is given by ep,q(x ) = dist(p, x ) + dist(x, q)- dist (p, q ). Given p E X and d > O, the d-excess of X at p is ep a = max rain e,q(x) xed(p,d)qes(p,d) e, whered(p,d) = {x E X s.t. dist(x,p) <_ d}ands(p,d) = {x E X s.t. dist(x,p) = d}. The d-excess of X is ed(x) = max e(. pex t, The excess of X is given by and the upper excess of X is (cf [AG], [GP1], and [02]). e(x) = rain max ep q(x), (p,q)exxx xex ' E (X ) - max min max ep q(x ). pcx qex xex ' Theorem 4 Let n E N, k E ~, and d > O. There is an e > 0 so that if M is a closed Riemannian n-manifold with Ricci M >_n- 1, sec M > k, ed(m) < e, and Diam M >_ 7r - e, then M is diffeomorphic to S n. Moreover, lime-.odistl(m, S ~) = O.

3 On radius, systole, and positive Ricci curvature 599 This generalizes Theorem 4 in [O2], Corollary 7 in [Wm], and a theorem in [Wu]. A key ingredient in the proofs of Theorems 1 and 4 is the following topological sphere theorem which was recently proven independently by Colding and Perelman. (See [Co] or [P].) Theorem 5 Given any n 6 N and k E IR there is an c > 0 so that if M is a closed Riemannian n-manifold with Ricci M > n - 1, sec M >_ k, and Diam M > 7r- c, then M is homeomorphic to S ~. For the proof of Theorem 1 we first combine Gromov's Precompactness Theorem (see [Gm 1 ]) with Yamaguchi's Fibration Theorem (see [Y2]) and conclude that it is sufficient to prove that any sequence of closed Riemannian n-manifolds which satisfies Ricci Mi ~ n -- 1, sec Mi >_ k, and (5) Radius Mi --+ 7r converges to S n in the Gromov-Hausdorff topology. By appealing to the proof of Theorem 3 in [O] (cf also the proof of Theorem 9 in [Wm]) we see that it is sufficient to show that any limit of such a sequence is n-dimensional. We will give two proofs of this. This seems justified since both are short, but rely on previous results whose proofs are quite long. The first proof has the defect that it fails to work in dimensions 3,7, and 15. For the first proof, we let Moo be the limit of a sequence which satisfies (5), and observe that we can use the proof of Lemma 1 in [GPI] to show that E(M~) = O. The Main Theorem and Proposition 22 in [Wm] then imply that Moo is homeomorphic to a sphere and that for all but finitely many i there is a fibration fi : Mi, Moo with connected fiber. By Theorem 5, all but finitely many of the Mi's are homeomorphic to S n. So by the commentary following Theorem 5.1 in [Br], we see that dim Moo must be n, unless n = 3,7, or 15. Throughout the remainder of the paper we adopt the convention that if p is a point in a metric space X and r > 0, then B(p,r) = {x C X s.t. dist(x,p) < r}. The idea behind the second proof is to use the argument in [P] to obtain a uniform lower bound for the filling radii of the Mi's and then to appeal to Gromov's famous (but difficult to prove) inequality, FillRad M <_ C(n)(vol M)~, where C(n) is a constant which depends only on n. (See [Gm2] for the definition of filling radius and the proof of this inequality.) For each point xi 6 Mi let A(xi) C Mi denote the set of points at maximal distance from xi, and let li C Mi denote the set of points Pi E Mi so that

4 600 F. Wilhelm Pi E A(qi) for some qi E Mi. The argument in [P] shows that the distance function from each point Pi C I i has no critical points in B(pi, Radius Mi), and it is elementary to show that E(Moo) = 0 implies/co = M~. We therefore obtain a uniform lower bound for the filling radii of the Mi 'S by appealing to the following modification of a result of Gromov. Lemma 6 Let M be a closed orientable Riemannian n-manifold. Suppose there is a sequence of numbers 0 < ~1 < ~2 <... < cn+l so that for every x C M and 2 every i <_ n, B (x, E i + ~ E'I ) can be contracted inside of B (x, Ci+l). Then 1 FillRad M >_ =el. L5 This can be proved by the retraction method in [Gm2], 4.5A, B, and C. (Cf also [Gm2], 1.2B and C and the proof of Theorem 2 in [K].) We will omit the proof of Corollary 2 since it is almost identical to the proof of Theorem 10 in [Wm]. (The only difference is that references to Theorem 1 are substituted for references to other less general radius sphere theorems.) The proof of Theorem 4 is also via the Gromov-Hausdorff convergence technique. We will show that any sequence of closed, Riemannian n-manifolds, {Mi }, which satisfies Ricci Mi >_ n - l, secmi > k (6) Diam Mi -+ ~, and ed(mi) ~ 0 converges to S" in the Gromov-Hausdorff topology, and then appeal once more to Gromov's Precompactness Theorem and Yamaguchi's Fibration Theorem. (See [Gml] and [Y2].) By Corollary 6 in [Win], any limit Moo of such a sequence is an almost Riemannian space with positive injectivity radius. Moreover, for all but finitely many i, there is an almost Riemannian fibration ~ : mi ----* Moo with connected fiber. (See [Wm] for the definition of almost Riemannian space and almost Riemannian submersion. Cf also Corollary 4.21 in [Y3].) The next step is to show that M~ is n-dimensional.'we will provide two ways of doing this. For the first, use Lemma 1 in [GP1] to conclude that e(m~) = 0. Then since ca(m) is also 0, it follows that Moo is homeomorphic to a sphere. By Theorem 5, all but finitely many of the Mi's are also topological spheres. So by appealing to [Br] we get dim Moo = n. (Unless n = 3, 7, or 15.) For the second proof, we first appeal to the argument in [P] to see that if Pi,qi E Mi satisfy dist(pi,qi) = diameter Mi and i is sufficiently large, then there are no critical points for the distance function from Pi other than qi. In particular, B(pi, -~) is contractible. (See [Ch] or [Gr] for a discussion of critical points of distance functions.)

5 On radius, systole, and positive Ricci curvature 601 By passing to a subsequence if necessary, we may assume that {Pi} and {qi} converge to points Poo,qoo C Moo. By Lemma 1 in [GP1], epo~,qr r ~ O. In particular, B(poo, r) is contractible for all r < 7r. For i sufficiently large we have fi-lb(p~, 4) C B(pi,-~) C fi-lb(p~, ~-). This is a contradiction because the inclusion map t : f.-ib(p~,88 > fi (P~, is a homotopy equivalence of spaces with nontrivial, reduced Z2- homology which factors through the contractible space B(pi, 2)" So dim Moo = n. Finally we can argue as in the proof of Theorem 9 in [Wm] (p. 1136) that vol Moo = vol S ~ and hence that vol Mi --~ vol S ~. The result follows from the main theorem in [YI]. Remarks: The argument in the second proof of Theorem 4 can also be used to give a third proof of Theorem 1. All of the above references to the main theorem in [Wm] can be replaced with references to Remark 4.20 in [Y3]. Acknowledgement. It is my pleasure to thank Grisha Perelman for several enlightening conversations about this paper. 1 woud especially like to thank him for suggesting the final formulation of Lemma 6, and for pointing out all of the main ideas in the second proof of Theorem 4. References [A] [AG] [Br] [Co] [Ch] [Cn] [Gml] [Gm2] [Gr] [GPll [GP2] [K] [M] [01] [o21 [P] M. Anderson, Metrics of positive Ricci curvature with large diameter, Manuscripta Math. 68, (1990), U. Abresch and D. Gromoll, On complete mani[olds qf nonnegative Ricci curvature, JAMS 3, (1990), W. Browder, Higher torsion in h-spaces, TAMS 108, (1963), T. Colding, Ricci diameter sphere theorem, in preparation. J. Cheeger, Critical points of distance.[unctions and applications to geometry, Geometric Topology: Recent developments, Montecatini Term6, 1990 (Ed. P. de Bartolomeis ans F. Tricerri) Springer Lecture Notes 1504 (1991), S. Y. Cheng, Eigenvalue comparison theorem and geometric applications, Math Z. 143, (1975), M. Gromov, Groups of polynomial growth and expanding maps, Publ. Math. I.H.E.S. 53, (1981), M. Gromov, Filling Riemannian manifidds, J. Differential Geometry, 18 (1983), K. Grove, Critical point theory for distance functions, Proc. A.M.S. Summer Institute on Differential Geometry, UCLA, K. Grove and P. Petersen, A pinching theoremjor homotopy sphere's, JAMS 3, (1990), K. Grove and P. Petersen, Volume comparison h la Alexandrov, Acta. Math., 169, (1992), M. Katz, The Filling radius of two-point homogeneous spaces, J. Differential Geometry, 18, (1983), S. B. Meyers, Riemannian man!fblds with positive mean curvature, Duke Math. J. 8, (1941), Y. Otsu, On man!fblds of positive Ricci curvature with large diameters, Math. Z. 206, (1991), Y. Otsu, On manijolds of'small excess, Amer. J. Math., to appear. G. Perelman, A diameter sphere theorem for manifolds of positive Ricci curvature. Math. Z., 218, (1995),

6 602 F. Wilhelm [PSZ] P. Petersen, Z. Shen, and S. Zhu, Manifolds with small excess and bounded curvature, Math. Z. 212, (1993), [SY] K. Shiohama and T. Yamaguchi, Positively curved manifolds with restricted diameters, Geometry of Manifolds (K. Shiohama, ed.), Prospectives in Math. 8, Academic Press, 1989, [Wm] F. Wilhelm, Collapsing to almost Riemannian spaces, Indiana Univ. Math. J. 41 (1992), [Wu] J-Y. Wu, Hausdorffconvergence and sphere theorems, Proc. of Symposia in Pure Mathematics UCLA [Y1] T. Yamaguchi, Lipshitz convergence of mani.[olds t~f positive Ricci curvature with large volume, Math. Ann. 284 (1989), [Y2] T. Yamaguchi, Collapsing and pinching under a lower curvature bound, Ann. of Math., 133, (1991), [Y3] T. Yamaguchi, A convergence theorem in the geometry of Alexandrov space, preprint. This article was processed by the author using the 16TEX style file pljourlm from Springer-Verlag.