Non-free isometric immersions of Riemannian manifolds

LOI, ANDREA;D'AMBRA G.

2007-01-01

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Abstract

Let (V , g) be a Riemannian manifold and let D be the isometric immersion opera- torwhich,toamap f:(V,g)→Rq,associatestheinducedmetricD(f)=g= f∗(⟨·,·⟩)on V , where ⟨·, ·⟩ denotes the Euclidean scalar product in Rq . By Nash–Gromov implicit func- tion theorem D is infinitesimally invertible over the space of free maps. In this paper we study non-free isometric immersions R2 → R4. We show that the operator D: C∞(R2, R4) → {G} (where {G} denotes the space of C∞- smooth quadratic forms on R2) is infinitesimally invert- ible over a non-empty open subset of A ⊂ C∞(R2,R4) and therefore D: A → {G} is an open map in the respective fine topologies.