An explicit formula for the Laplace-Beltrami operator on the Stiefel manifold
Petre Birtea, Ioan Casu, Dan Comanescu
TL;DR
This work derives an explicit ambient-coordinate formula for the Laplace-Beltrami operator on the orthogonal Stiefel manifold $St_p^n$, viewing it as a constraint submanifold of ${\mathcal M}_{n\times p}({\mathbb R})$ endowed with the Frobenius metric. By applying a general constraint-manifold framework, the authors obtain a closed expression $\Delta_{St_p^n}\tilde{f}(U)=\Delta f(U)-\left(n-\frac{p+1}{2}\right)\mathrm{tr}(U^T \nabla f(U))-\frac{1}{2}\mathrm{tr}\left((\mathbb{I}_p \otimes (UU^T)+\Lambda(U)) \mathrm{Hess} f(U)\right)$, with Lagrange multiplier terms encoded in $\Sigma(U)$ and $\sigma_\alpha(U)$. The result recovers known formulas for the sphere ($p=1$) and $SO(n)$ ($p=n$) and aligns with projection-based approaches, yielding a practical, coordinate-based tool for analysis on constraint manifolds such as $St_p^n$. The paper also clarifies the ambient-geometry structure via the tangent-space decomposition and the role of the matrix $\Lambda(U)$ (and its relation to the commutation matrix).
Abstract
We derive an explicit formula for the Laplace-Beltrami operator on the orthogonal Stiefel manifold, viewed as a constraint submanifold of the Euclidean space of real matrices equipped with the Frobenius metric. Using the general framework of Laplace operators on constraint manifolds, we provide the formula for the Laplace-Beltrami operator in terms of the ambient Euclidean coordinates. The result extends previously known cases, recovering the formulas for the sphere and the special orthogonal group as particular instances.