Partially Isometric Immersions and Free Maps

In this paper we investigate the existence of “partially ” isometric im-mersions. These are maps f: M → Rq which, for a given Riemannian manifold M, are isometries on some sub-bundle H ⊂ 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 di-mension 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 R2, Lagrangian distributions of completely integrable systems, Hamiltonian distributions of a particular kind of Poisson Bracket