Central splitting of manifolds with no conjugate points

TL;DR

This paper explores conditions under which certain functions on compact manifolds with no conjugate points lead to manifold splitting, generalizing classical splitting theorems for nonpositive curvature.

Contribution

It introduces a new family of functions related to central Busemann functions that characterize manifold splitting in the absence of conjugate points.

Findings

Functions vanish when all central Busemann functions are sub- or superharmonic

Central Busemann functions are totally geodesic if convex or concave

Generalizes splitting theorems for manifolds with no focal points and nonpositive curvature

Abstract

Each compact Riemannian manifold with no conjugate points admits a family of functions whose integrals vanish exactly when central Busemann functions split linearly. These functions vanish when all central Busemann functions are sub- or superharmonic. When central Busemann functions are convex or concave, they must be totally geodesic. These yield generalizations of the splitting theorems of O'Sullivan and Eberlein for manifolds with no focal points and, respectively, nonpositive curvature.