Transverse totally geodesic submanifolds of the tangent bundle

Abstract

It is well-known that ifis a smooth vector field on a given Rie- mannian manifold M n thennaturally defines a submanifold �(M n ) transverse to the fibers of the tangent bundle TM n with Sasaki metric. In this paper, we are interested in transverse totally geodesic subman- ifolds of the tangent bundle. We show that a transverse submanifold N l of TM n (1 ≤ l ≤ n) can be realized locally as the image of a sub- manifold F l of M n under some vector fielddefined along F l . For such images �(F l ), the conditions to be totally geodesic are presented. We show that these conditions are not so rigid as in the case of l = n, and we treat several special cases (� of constant length, � normal to F l , M n of constant curvature, M n a Lie group anda left invariant vector field).