Partially Isometric Immersions and Free Maps

Details Partially Isometric Immersions and Free Maps Partially Isometric Immersions and Free Maps

2010-07-18

arxiv.org/abs/1007.3024v1

Mathematics

Differential Geometry

19 pages, 1 figure

Scientific paper

In this paper we investigate the existence of ``partially'' isometric immersions. These are maps f:M->R^q which, for a given Riemannian manifold M, are isometries on some sub-bundle H of TM. The concept of free maps, which is essential in the Nash--Gromov theory of isometric immersions, is replaced here by that of H-free maps, i.e. maps whose restriction to H is free. We prove, under suitable conditions on the dimension q of the Euclidean space, that H-free maps are generic and we provide, for the smallest possible value of q, explicit expressions for H-free maps in the following three settings: 1-dimensional distributions in R^2, Lagrangian distributions of completely integrable systems, Hamiltonian distributions of a particular kind of Poisson Bracket.

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