A generalization of parallelism in Riemannian geometry, the $C\sp{\omega }$ 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 M be a (not necessarily isometric) immersion of Riemannian manifolds. Let G: T(N) 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. 0. Introduction. The geodesics in a Riemannian manifold M, of dimension m, can be described in two independent ways: they are the critical points for the variational problem for minimal 1-dimensional area with fixed endpoints and they are also the auto-parallels, the curves whose tangent vector fields are parallel. More generally, the critical points for the variational problem for minimal p dimensional area, 1 _p <im, with fixed boundary are the minimal immersions. Received by the editors December 30, 1969. AMS 1969 subject classifications. Primary 5372, 5304, 5350, 5370, 5374; Secondary 3503, 3504, 3596, 5730.