Laplace-Beltrami operator on the orthogonal group in Cartesian coordinates

TL;DR

This paper derives an explicit formula for the Laplace-Beltrami operator on the orthogonal group using ambient Euclidean coordinates, facilitating computations on this constraint manifold.

Contribution

It provides a general formula for the Laplace-Beltrami operator on constraint manifolds, specifically applied to the orthogonal group in Cartesian coordinates.

Findings

Explicit formula for Laplace-Beltrami on orthogonal group

Application to relevant functions on the group

Utilizes embedded gradient vector field method

Abstract

Using the embedded gradient vector field method (see P. Birtea, D. Comanescu, Hessian operators on constraint manifolds, J. Nonlinear Science 25, 2015), we present a general formula for the Laplace-Beltrami operator defined on a constraint manifold, written in the ambient coordinates. Regarding the orthogonal group as a constraint submanifold of the Euclidean space of $n\times n$ matrices, we give an explicit formula for the Laplace-Beltrami operator on the orthogonal group using the ambient Euclidean coordinates. We apply this new formula for some relevant functions.