Oh right, this is supposed to be a math blog, isn’t it?

So, I was trying to understand the proof in do Carmo (which is a great book, incidentally) of Synge-Weinstein, and I tried to write an expository paper on it.  Unfortunately I can’t get the formatting perfect in WordPress since some of the equations go out of the margins, but most if it should look ok.   I thus  made a PDF of this too.  I dressed it up in formal article formatting just for the heck of it.

We give an exposition of the proof of a few results in global Riemannian geometry due to Synge and Weinstein using variations of the energy integral.

1. Introduction

One of the big refrains of modern Riemannian geometry is that curvature determines topology. Recall, for instance, the basic Cartan-Hadamard theorem that a complete, simply connected Riemannian manifold of nonpositive curvature is diffeomorphic to {\mathbb{R}^n} under the exponential map. We proved this basically by showing that {\exp_p} is nonsingular under the hypothesis of nonnegative curvature (using Jacobi fields) and that it was thus a covering map (the latter part was relatively easy). More difficult, and relevant to the present topic, was the Bonnet-Myers theorem, which asserted the compactness of a complete Riemannian manifold with bounded-below, positive Ricci curvature. The proof there showed that a long enough geodesic could not minimize energy (by using the second variation formula—recall that the second variation formula is intimately connected with curvature), and therefore could not minimize length. Since the distance between two points in a complete Riemanninan manifold is the length of the shortest geodesic between them (Hopf-Rinow!), this implied a bound on the diameter.

Today, however, we’re going to assume at the outset that the manifold in question is already compact. One of the theorems will be that a compact, even-dimensional orientable manifold of positive curvature is simply connected. In particular, there is no metric of everywhere positive sectional curvature on the torus {\mathbb{T}^2}. (more…)