Non-free isometric immersions of Riemannian manifolds

摘要

Let (V, g) be a Riemannian manifold and let D be the isometric immersion operator which, to a map f : (V,g) → R~q, associates the induced metric D(f) = g = f~* (<.,.>) on V, where <.,.> denotes the Euclidean scalar product in R~q. By Nash–Gromov implicit function theorem D is infinitesimally invertible over the space of free maps. In this paper we study non-free isometric immersions R~2 → R~4. We show that the operator D:C~∞(R~2,R~4) → {G} (where {G} denotes the space of C~∞- smooth quadratic forms on R~2) is infinitesimally invertible over a non-empty open subset of A is contained in C~∞(R~2, R~4) and therefore D: A → {G} is an open map in the respective fine topologies.