Laplace Transformations of Submanifolds

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.