On Two Natural Riemannian Metrics on a Tube

Abstract

During an operation of surgery on a Riemannian manifold and along a given embedded submanifold, one needs to replace the (old) metric induced by the exponential map on a tubular neighborhood of the submanifold by the Sasakian metric. So a good understanding of the behavior of these two metrics is important, this is our main goal in this paper. In particular, we prove that these two metrics are tangent up to the order one if and only if the submanifold is totally geodesic. In the case where the ambient space is an Euclidean space, we prove that the difference of these two metrics is quadratic in the radius of the tube and depends only on the second fundamental form of the submanifold. Also the case of spherical and hyperbolic space forms are studied.