Extensions of the Rauch comparison theorem to submanifolds
Frank W. Warner · Transactions of the American Mathematical Society · 1966
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 [2] for the case in which the submanifolds are themselves geodesies.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.In §2 we review some of the basic geometry of submanifolds and give a precise statement of the Rauch comparison theorem.In §3 we give a formal setup and proof of the comparison theorem which we then apply to submanifolds in §4.2. Preliminaries.We refer the reader to [1] or [3] for details and proofs of the material summarized in this section.Unless we specify otherwise, we assume all manifolds and maps to be differentiable of class C00.By a differentiable map with domain a closed interval [a, fe] of the real line we mean one which can be extended to be differentiable on an open neighborhood of the interval.A piecewise differentiable map on [a,b] is one for which there is a partition a = a0 < a] < •■■ < a" = fe of [a,fe] such that the map is differentiable on each [a¡,a¡+1].Let M be a d-dimensional Riemannian manifold with d -2.We denote the