Laplace-Beltrami operator on the orthogonal group in ambient (Euclidean) coordinates

Using the embedded gradient vector field method (see P. Birtea, D. Comănescu (2015) [7]), 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.