The geometry of sectional curvatures

Abstract

A large class of questions in differential geometry involves the relationship between the geometry and the topology of a Riemannian (= positive-definite) manifold. We briefly review the status of the following question from this class: given that a compact, even-dimensional manifold admits a Riemannian metric of positive sectional curvatures, what can one say about its topology? Very few manifolds are known to admit such metrics. For example, is it not known whether or not the product of then-sphere with itself (n ≥ 2) does. One answer to the question above is provided by Synge's theorem: if the manifold is orientable, then it is simply connected. Another possible answer is given by the Hopf conjecture: such a manifold necessarily has positive Euler number. The Hopf conjecture is known to be true for homogeneous manifolds, and for arbitrary manifolds in dimensions two and four. This last result has two, apparently entirely different, proofs, one using Synge's theorem and the other the Gauss-Bonnet formula. Neither, it is shown, can be generalized directly to dimensions six or greater. The Hopf conjecture in these higher dimensions remains open.