  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.
