Totally geodesic maps into manifolds with no focal points

TL;DR

This paper proves new topological and geometric properties of totally geodesic maps into manifolds with no focal points, extending classical results without using geometric flows or Bochner identities.

Contribution

It establishes the path-connectedness and energy-minimizing nature of totally geodesic maps into manifolds with no focal points, generalizing splitting theorems and non-collapsing results.

Findings

Set of totally geodesic maps is path-connected.

Such maps are energy-minimizing when nonempty.

Results extend classical theorems to no focal points setting.

Abstract

The set of totally geodesic representatives of a homotopy class of maps from a compact Riemannian manifold $M$ with nonnegative Ricci curvature into a complete Riemannian manifold $N$ with no focal points is path-connected and, when nonempty, equal to the set of energy-minimizing maps in that class. When $N$ is compact, each map from a product $W \times M$ into $N$ is homotopic to a map that's totally geodesic on each $M$-fiber. These results may be used to extend to the case of no focal points a number of splitting theorems of Cao-Cheeger-Rong about manifolds with nonpositive sectional curvature and, in turn, to generalize a non-collapsing theorem of Heintze-Margulis. In contrast with previous approaches, they are proved using neither a geometric flow nor the Bochner identity for harmonic maps.