Partially Isometric Immersions and Free Maps

TL;DR

This paper explores the existence and genericity of H-free maps, a generalization of free maps, for partially isometric immersions in various geometric contexts, providing explicit constructions in specific cases.

Contribution

It introduces the concept of H-free maps, proves their genericity under certain conditions, and constructs explicit examples in three geometric settings.

Findings

H-free maps are generic under suitable dimension conditions.

Explicit H-free maps are constructed for specific distributions.

The concept extends the Nash--Gromov theory to partial isometries.

Abstract

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.