The Ricci Curvature
In this chapter we deal with problems concerning Ricci Curvature mainly: Prescribing the Ricci curvature Ricci curvature with a given sign Existence of Einstein metrics.
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The Ricci Curvature
§1. About the Different Types of Curvature 9.1 In this chapter we deal with problems concerning Ricci Curvature mainly: -
Prescribing the Ricci curvature Ricci curvature with a given sign Existence of Einstein metrics.
This latter problem: to decide if a Riemannian manifold carries an Einstein metric, will be one of the important questions in Riemannian geometry for the next decades. Indeed, in spite of recent results that we will talk about, Ricci curvature is not yet well understood. Ricci curvature lies between sectional and scalar curvatures. We saw that scalar curvature is now well-known and we recall below some results concerning sectional curvature. In this chapter (except in §1.4) we suppose that the dimension of the manifold is greater than 2. 1.1 The Sectional Curvature
9.2 We will mention some well-known results which prove that it is a strong property for a manifold to have its sectional curvature of a given sign. We see that it is impossible for a manifold to carry two metrics, the sectional curvatures of which are of opposite sign. Theorem 9.2. A complete connected Riemaniann manifold (Mn , g) has constant sectional curvature if and only if it is isometric to Sn, ]Rn or Hn the hyperbolic space, or one of their quotients by a group r of isometries which acts freely and properly. Sn, ]Rn and Hn are endowed with their canonical metrics. 9.3 Theorem (Synge). A compact connected orientable Riemannian manifold of even dimension with strictly positive sectional curvature is simply connected. The proof is by contradiction. If the manifold is not simply connected there is a shortest closed geodesic r in any nontrivial homotopy class. As the manifold is orientable and of even dimension, there exists a unit parallel vector field along r orthogonal to r. Then we can consider the second variation of the
T. Aubin, Some Nonlinear Problems in Riemannian Geometry © Springer-Verlag Berlin Heidelberg 1998
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9. The Ricci Curvature
length integral in the direction of this vector field (related to a family r A of closed curves near ro = as we do for the proof of Myers' Theorem 1.43. The hypothesis on the sign of the sectional curvature implies that this second variation is negative, which is a contradiction, since r would not be the shortest curve in its homotopy class.
n,
9.4 Theorem. A complete simply-connected Riemannian manifold (M, g) with nonpositive sectional curvature is diffeomorphic to ]Rn. Proof. Let P be any point of M. P has no conjugate point (Theorem 1.48), so exp p is a diffeomorphism from ]Rn to M (Theorem 1.46).
Corollary 9.4. A compact Riemannian manifold (M, g) with non-positive sectional curvature cannot carry a metric y with positive Ricci curvature. Indeed by Myers' Theorem 9.6, if Y exists, (M, g) has a compact universal covering space (M,7r*Y), 7r : M - t M. This is in contradiction with Theorem 9.4 which asserts that the universal covering space of (M, g) is diffeomorphic to ]Rn. 1.2. The Scalar Curvature 9.S In this section we suppose n ~ 3. We saw that there are
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