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Article Dans Une Revue Anal.Appl.(Singap) Année : 2006

$C\sp\infty$-regularity of a manifold as a function of its metric tensor.

Cristinel Mardare
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Résumé

A basic theorem from differential geometry asserts that if the Riemann curvature tensor associated with a smooth field C of positive-definite symmetric matrices of order n vanishes in a simply-connected open subset Ω of R^n, then C is the metric tensor of a manifold isometrically immersed in R^n. If Ω is connected, then the isometric immersion Θ defined in this fashion is unique up to isometries of R^n. We prove that, if the set Ω is bounded and has a smooth boundary, then the mapping C → Θ is of class C^∞ between manifolds in appropriate Banach spaces.
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Dates et versions

hal-00112617 , version 1 (09-11-2006)

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Cristinel Mardare. $C\sp\infty$-regularity of a manifold as a function of its metric tensor.. Anal.Appl.(Singap), 2006, 4, pp.19-30. ⟨10.1142/S0219530506000681⟩. ⟨hal-00112617⟩
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