  There are two natural Chern-Simons theories associated with the embedding of
a three-dimensional surface in Euclidean space; one is constructed using the
induced metric connection -- it involves only the intrinsic geometry, the other
is extrinsic and uses the connection associated with the gauging of normal
rotations. As such, the two theories appear to describe very different aspects
of the surface geometry. Remarkably, at a classical level, they are equivalent.
In particular, it will be shown that their stress tensors differ only by a null
contribution. Their Euler-Lagrange equations provide identical constraints on
the normal curvature. A new identity for the Cotton tensor is associated with
the triviality of the Chern-Simons theory for embedded hypersurfaces implied by
this equivalence. The corresponding null surface stress capturing this
information will be constructed explicitly.
