A Short Survey on Curvature and Topology
We have now covered almost half of the chapters of the present textbook and the more elementary aspects of the subject. Before penetrating into more advanced topics, a short survey on some directions of global Riemannian geometry may be a useful orientati
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We have now covered almost half of the chapters of the present textbook and the more elementary aspects of the subject. Before penetrating into more advanced topics, a short survey on some directions of global Riemannian geometry may be a useful orientation guide. Because of the size and scope of the present book, this survey needs to be selective. A ~asic question, formulated in particular by H. Hopf, is to what extent the existence of a Riemannian metric with particular curvature properties restricts the topology of the underlying differentiable manifold. The classical example is the Gauss-Bonnet Theorem. Let M be a compact two-dimensional Riemannian manifold with curv~.&ture K. Then its Euler characteristic is determined by
x(M)
= 2~
I
K dvol M.
M
We have also seen some higher dimensional examples already, namely the Theorem 4.1.2 of Synge on manifolds with positive sectional 011rvature, the Theorem 3.5.1 of Bochner and the Bonnet-Myers Theorem (Corollary 4.3.1) on manifolds of positive Ricci curvature. In chapter 8, we shall present results for nonpositive sectional curvature, namely the Hadamard-Cartan Theorem (Corollary 8.7.1) that a simply-connected, complete manifold of nonpositive sectional curvature is diffeomorphic to some r, and the Preissmann Theorem (Corollary 8.7.5) that any abelian subgroup of the fundamental group of a compact manifold of negative sectional curvature is infinite cyclic, i.e. isomorphic to Z. In order to put these results in a better perspective, we want to discuss the known implications of curvature properties for the topology more systematically. We start with the implications of positive sectional curvature. Here, we have the
Sphere Theorem. Let M be a compact, simply connected Riemannian manifold whose sectional curvature K satisfies 1 0 < 4~~:< K ~ ~~:
J. Jost, Riemannian Geometry and Geometric Analysis © Springer-Verlag Berlin Heidelberg 1998
204
A Short Survey on Curvature and Topology
for some fixed number dimM).
K..
Then M is homeomorphic to the sphere S"' (n
=
This was shown by M. Berger, Les varietes Riemanniennes (t)-pincees, Ann. Scuola Norm. Sup. Pisa, Ser. III, 14 (1960), 161-170, and W. Klingenberg, Uber Riemannsche Mannigfaltigkeiten mit positiver Kriimmung, Comm. Math. Helv. 35 {1961), 47-54. The pinching number t is optimal in even dimensions ~ 4. Because crm (see §6.1) is simply connected, has sectional curvature between and 1 for its Fubini-Study metric and is not homeomorphic to S2m for m > 1. In odd as shown by U. dimesnions, the pinching number can be decreased below Abresch and W. Meyer, Pinching below injectivity radius, and conjugate radius, J.Diff.Geom. 40, {1994) 643-691, but the optimal value of the pinching constant is unkonown at present. For n = 2 or 3, the conclusion is valid already if M has positive sectional curvature. For n = 2, this follows from the Gauf.l..Bonnet Theorem. For n = 3, R. Hamilton, Three-manifolds with positive Ricci curvature, J. Diff. Geom. 17 (1982), 255-306, showed that any simply connected compact manifold of
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