TY - JOUR
TI - Totally geodesic maps into manifolds with no focal points
DO - https://doi.org/doi:10.7282/T36Q1VQB
PY - 2014
AB - The space of totally geodesic maps in each homotopy class [F] from a compact Riemannian manifold M with non-negative Ricci curvature into a complete Riemannian manifold N with no focal points is path-connected. If [F] contains a totally geodesic map, then each map in [F] is energy-minimizing if and only if it is totally geodesic. When N is compact, each map from a product W x M into N is homotopic to a smooth map that's totally geodesic on the M-fibers. These results generalize the classical theorems of Eells-Sampson and Hartman about manifolds with non-positive sectional curvature and are proved using neither a geometric flow nor the Bochner identity. They can be used to extend to the case of no focal points a number of splitting theorems proved by Cao-Cheeger-Rong about manifolds with non-positive sectional curvature and, in turn, to generalize a theorem of Heintze-Margulis about collapsing. The results actually require only an isometric splitting of the universal covering space of M and other topological properties that, by the Cheeger-Gromoll splitting theorem, hold when M has non-negative Ricci curvature. The flat torus theorem is combined with a theorem about the loop space of a manifold with no conjugate points to show that the space of totally geodesic maps in [F] is path-connected. A center-of-mass method due to Cao-Cheeger-Rong is used to construct a homotopy to a totally geodesic map when M is compact. The asymptotic norm of a Z^m-equivariant metric is used to show that the energies of C^1 maps in [F] are bounded below by a constant involving the energy of an affine surjection from a flat Riemannian torus onto a flat semi-Finsler torus, with equality for a given map if and only if it is totally geodesic. This builds on work of Croke-Fathi. It is also shown that the ratio of convexity radius to injectivity radius can be made arbitrarily small over the class of compact Riemannian manifolds of any fixed dimension at least two. This uses Gulliver's examples of manifolds with focal points but no conjugate points.
KW - Mathematics
KW - Geodesics (Mathematics)
KW - Geometry, Riemannian
LA - eng
ER -