A generalization of parallelism in Riemannian geometry, the $C^{\omega }$ case

Abstract:

The concept of parallelism along a curve in a Riemannian manifold is generalized to parallelism along higher dimensional immersed submanifolds in such a way that the minimal immersions are self parallel and hence correspond to geodesics. Let $g:N \to M$ be a (not necessarily isometric) immersion of Riemannian manifolds. Let $G:T(N) \to T(M)$ be a tangent bundle isometry along $g$, that is, $G$ covers $g$ and maps fibers isometrically. By mimicing the construction used for isometric immersions, it is possible to define the mean curvature vector field of $G.G$ is said to be parallel along $g$ if this vector field vanishes identically. In particular, minimal immersions have parallel tangent maps. For curves, it is shown that the present definition reduces to the definition of Levi-Civita. The major effort is directed toward generalizations, in the real analytic case, of the two basic theorems for parallelism. On the one hand, the existence and uniqueness theorem for a geodesic in terms of data at a point extends to the well-known existence and uniqueness of a minimal immersion in terms of data along a codimension one submanifold. On the other hand, the existence and uniqueness theorem for a parallel unit vector field along a curve in terms of data at a point extends to a local existence and uniqueness theorem for a parallel tangent bundle isometry in terms of mixed initial and partial data. Since both extensions depend on the Cartan-Kahler Theorem, a procedure is developed to handle both proofs in a uniform manner using fiber bundle techniques.