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      Laplace Transformations of Submanifolds

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          Abstract

          Let \(x : M \to E^m\) be an isometric immersion of a Riemannian manifold \(M\) into a Euclidean \(m\)-space. Denote by \(\Delta\) the Laplace operator of \(M\). Then \(\Delta\) gives rise to a differentiable map \(L :M \to E^m\), called the Laplace map, defined by \(L(p)=(\Delta x)(p)\), \(p\in M\). We call \(L(M)\) the Laplace image, and the transformation \(L :M \to L(M)\) from \(M\) onto its Laplace image \(L(M)\) the {\it Laplace transformation}. In this monograph, we provide a fundamental study of the Laplace transformations of Euclidean submanifolds.

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          Most cited references 29

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          On surfaces of finite type in Euclidean $3$-space

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            Quadrics of finite type

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              An extension of Takahashi's theorem

               OscarJ. Garay (1990)
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                Author and article information

                Journal
                04 July 2013
                Article
                1307.1515

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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                Differential Geometry
                126 pages. Published by the Center for Pure and Applied Differential Geometry (Leuven and Brussel, Belgium), 1995
                math.DG

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