Extensions of the Rauch comparison theorem to submanifolds

Abstract

The Rauch comparison theorem yields a metric comparison of the lengths of Jacobi fields along geodesies in different Riemannian manifolds under suitable initial conditions and suitable hypotheses on the curvatures and on the nonexistence of conjugate points. Part of the initial condition is that the Jacobi fields should vanish at the initial points. In this paper we show how Rauch's theorem and proof extend to Jacobi fields satisfying more general initial conditions; namely, to Jacobi fields associated with submanifolds. Berger has given such an extension in For more general submanifolds the initial conditions will involve the second fundamental forms ; and so in the comparison theorem, one needs an additional hypothesis comparing the second fundamental forms. Instead of conjugate points, one is now concerned with focal points. Several new factors enter. One is that one has to apply, in certain cases, some special boundary conditions in order to get a comparison, and another is that the comparison generally does not hold as far as the first focal point in contrast to the Rauch case where the comparison holds as far as the first conjugate point.