C1,α isometric embeddings of polar caps

Abstract

We study isometric embeddings of C2 Riemannian manifolds in the Euclidean space and we establish that the Hölder space C1,12 is critical in a suitable sense: in particular we prove that for α>12 the Levi-Civita connection of any isometric immersion is induced by the Euclidean connection, whereas for any α<12 we construct C1,α isometric embeddings of portions of the standard 2-dimensional sphere for which such property fails.