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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.

### Most cited references29

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

(1990)
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(1990)
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### An extension of Takahashi's theorem

(1990)
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### Author and article information

###### Journal
04 July 2013
###### Article
1307.1515