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\(C^{1, \alpha }\) isometric embeddings of polar caps. (English) Zbl 1436.53006

Summary: We study isometric embeddings of \(C^2\) Riemannian manifolds in the Euclidean space and we establish that the Hölder space \(C^{1 , \frac{ 1}{ 2}}\) is critical in a suitable sense: in particular we prove that for \(\alpha > \frac{ 1}{ 2}\) the Levi-Civita connection of any isometric immersion is induced by the Euclidean connection, whereas for any \(\alpha < \frac{ 1}{ 2}\) we construct \(C^{1 , \alpha}\) isometric embeddings of portions of the standard 2-dimensional sphere for which such property fails.

MSC:

53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
26B35 Special properties of functions of several variables, Hölder conditions, etc.
58D10 Spaces of embeddings and immersions
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
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