The isometry groups of manifolds admitting nonconstant convex functions (Q1087153)

The main result of this paper is the following theorem: If M is a complete Riemannian manifold with noncompact isometry group, and if there is a convex function \(\phi\) : \(M\to {\mathbb{R}}\) with \(\{x\in M| \quad \phi (x)=\inf_ M\phi \}=\emptyset\) and \(\{x\in M| \quad \phi (x)=\alpha \}\) compact for all \(\alpha\in {\mathbb{R}}\), then M is isometric to the product \(N\times {\mathbb{R}}\) of a compact \(C^{\infty}\) Riemannian manifold N and the real line \({\mathbb{R}}\). The elaborate argument proving this theorem makes use of two lemmas of independent interest, concerning also the properties of a complete Riemannian manifold having special convex functions.