Laplace Transformations of Submanifolds

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.