Partially isometric immersions and free maps

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

Partial immersions and partially free maps

“…Next theorem shows that, independently on the topology of H and of M, both sets are non-empty if q is big enough: [2].) The sets Imm H (M, R q ) and Free H (M, R q ) are dense in C ∞ (M, R q ) for, respectively, q m + k and q m + k + s k .…”

“…When H is Hamiltonian, the existence of such a function was proved by Weiner in its lemma in [10]. When H is of finite type, it was proved in Lemma 3.1 of [2]. 2 Example 3.…”

“…Thanks to Theorem 2, we can now reformulate Theorems 1.1-1.3 of [2] so that it is clear that they all depend on the existence of an H-immersion. Consider first the case of 1-distributions H in the plane (see Section 3.1 in [2]).…”

“…Consider first the case of 1-distributions H in the plane (see Section 3.1 in [2]). Kaplan proved [6] that all 1-distributions in the plane are orientable, so that there exists a vector field everywhere non-zero such that H = span ξ .…”

Partial Isometries of a Sub-Riemannian Manifold

“…Definition 2.1. A C 1 map from M to N is called an H-immersion if its derivative restricts to a monomorphism on H (We have borroed this terminology from [2]).…”