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Transactions of the American Mathematical Society

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

Author: Alan B. Poritz
Journal: Trans. Amer. Math. Soc. 152 (1970), 461-494
MSC: Primary 53.72
MathSciNet review: 0268813
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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.

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Keywords: Parallelism, auto-parallel, least-area variational problem, geodesic, minimal immersion, connection, Levi-Civita connection, vector bundle, Euclidean vector bundle map, tangent bundle isometry, Grassmann manifold, oriented Grassmann manifold, canonical vector bundle, bundle of adapted frames, isometric immersion, second fundamental form, mean curvature vector field, parallel vector field, parallel tangent bundle isometry, differential ideal, vector bundle valued form, connection form, equivariant form, regular integral plane, polar space, Cartan-Kahler Theorem, initial data problem, bundle of partial data
Article copyright: © Copyright 1970 American Mathematical Society