The Laplace-Beltrami operator is a generalization of the Laplacian to functions defined on Riemannian manifolds. It combines the concepts of the Laplacian and the metric structure of a manifold, allowing for the analysis of geometric properties and their relation to differential equations. This operator plays a crucial role in geometric analysis and is used in various applications including heat flow, harmonic forms, and the study of differential equations on curved spaces.
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