It is well-known that if ξ is a smooth vector field on a given Riemannian manifold Mn then ξ naturally defines a submanifold ξ(Mn)
Nearby in the stack
transverse to the fibers of the tangent bundle
TMn
with Sasaki metric. In this paper, we are interested in transverse totally geodesic submanifolds of the tangent bundle. We show that a transverse submanifold
Nl
of
TMn
(
1≤l≤n
) can be realized locally as the image of a submanifold
Fl
of
Mn
under some vector field
ξ
defined along
Fl
. For such images
ξ(Fl)
, the conditions to be totally geodesic are presented. We show that these conditions are not so rigid as in the case of