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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  ×  M    into   N   is homotopic to a map that is 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 noncollapsing theorem of Heintze–Margulis. In contrast with previous approaches, they are proved using neither a geometric flow nor the Bochner identity for harmonic  maps.

Totally geodesic maps into manifolds with no focal points