Existence of parallel sections of a vector bundle

Abstract

We consider when a smooth vector bundle endowed with a connection possesses non-trivial, local parallel sections. This is accomplished by means of a derived flag of subsets of the bundle. The procedure is algebraic and rests upon the Frobenius Theorem. For the case of the bundle of symmetric bilinear forms on a manifold our method answers the question as to when a connection on the manifold is locally a metric connection.