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Two natural metrics and their covariant derivatives on a manifold of embeddings

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Abstract

OnE(M, ℝn), the Fréchet-manifold of all smooth embeddings of a smooth, compact, closed, orientable manifoldM (of dimensionn-1) into ℝn two natural metricsG and\(\bar G\) are considered. The metric\(\bar G\) plays a central rôle in elasticity theory. Using a generalised notion of the Fréchet derivative their respective spraysS′ and\(\bar S'\) and the correspoonding Levi-Civita connections are computed. BothS′ and\(\bar S'\) are smooth in a well defined sense. In contrast toS′ the spray\(\bar S'\) turns out to be trivial.

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  1. Lehrstuhl für Mathematik I Fakultät für Mathematik und Informatik, Universität Mannheim, D-6800, Mannheim, Federal Republic of Germany

    E. Binz

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  1. E. Binz
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Binz, E. Two natural metrics and their covariant derivatives on a manifold of embeddings. Monatshefte für Mathematik 89, 275–288 (1980). https://doi.org/10.1007/BF01659491

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