The Laplace-Beltrami-Operator on Riemannian Manifolds

This report mainly illustrates a way to compute the Laplace-Beltrami-Operator on a Riemannian Manifold and gives information to why and where it is used in the Analysis of 3D Shapes. After a brief introduction, an overview over the necessary properties of manifolds for calculating the Laplacian is given. Furthermore the two operators needed for defining the Laplace-Beltrami-Operator the gradient and the divergence are introduced. Both of them are individually derived from the geometric meaning of the operator in Euclidean space, which then is translated into a Riemannian Manifold. With these operators defined, their formulas are combined into a specification used to calculate the Laplace-Beltrami-Operator. Finally, some examples for its applications are presented. Most of this report uses [1] as a reference, for this reason it won’t be specifically noted. 1 Why do we need the Laplace-Beltrami-Operator? Figure 1: 3D-model of a woman in different positions, coloured by using the 8th eigenfunction of the Laplace-Beltrami operator. The third pose is coloured inverse to the other two. As the title states, the main objective of this report is to illustrate the mathematical basics of how to calculate the Laplace Operator on Riemannian Manifolds. So what does this have to do with 3D-objects? Most of the time 3D-objects are represented as a surface, which (with some math behind it) can be approximated by a manifold e.g. with the help of the Sturm-Liouville’s decomposition [2]. Many methods for analyzing shapes use the Eigenfunctions of the LaplaceBeltrami-Operator (or the Laplacian for short) to calculate a metric bound to vertices. As shown in Figure 1 those Eigenfunctions are not changed by isometric deformations and therefore are a good way to specify a certain point on the object. Since the Laplace operator is defined as ∆ := div∇, (1)