  We argue that two dimensional classical SU(2) Yang-Mills theory describes the
embedding of Riemann surfaces in three dimensional curved manifolds.
Specifically, the Yang-Mills field strength tensor computes the Riemannian
curvature tensor of the ambient space in a thin neighborhood of the surface. In
this sense the two dimensional gauge theory then serves as a source of three
dimensional gravity. In particular, if the three dimensional manifold is flat
it corresponds to the vacuum of the Yang-Mills theory. This implies that all
solutions to the original Gauss-Codazzi surface equations determine two
dimensional integrable models with a SU(2) Lax pair. Furthermore, the three
dimensional SU(2) Chern-Simons theory describes the Hamiltonian dynamics of two
dimensional Riemann surfaces in a four dimensional flat space-time.
