Local and Global Geometries of the Bundles

Abstract

Let be a Riemannian manifold and its tangent bundle. If is a smooth vector field on then naturally defines a submanifold
transverse to the fibers. It is well known that all totally geodesic vector fields along submanifolds of generate
submanifolds in which are transverse to the fibers of . In this paper we study submanifolds of with which are
transverse to the fibers and any transverse submanifold of can be realized locally as the image of a submanifold of under
some vector field . We are interested in submanifolds among this class which are totally geodesic.