Non-free isometric immersions of Riemannian manifolds

Abstract

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