On the integrability conditions for extended objects

Abstract

The integrability conditions for the embedding of a d-dimensional world manifold generated by a (d-1)-dimensional extended object in a flat space are examined. It is shown that, for the case of non-minimal world manifolds with d>2, these conditions contain the Einstein and Yang-Mills equations as an independent set of equations, where the first and third fundamental forms act as gravitation and gauge fields respectively, while the second fundamental form plays the role of the source field. Using these results the string model of relativity is shown to be compatible with general relativity.