Chavel, Isaac 1984 , Eigenvalues in Riemannian Geometry, Pure and Applied Mathematics, 115 2nd ed | Geometrie des groupes de transformations |
---|---|
On functions, the Laplace—de Rham operator is actually the negative of the Laplace—Beltrami operator, as the conventional normalization of the assures that the Laplace—de Rham operator is formally , whereas the Laplace—Beltrami operator is typically negative | It is convenient to regard the sphere as isometrically embedded into R n as the unit sphere centred at the origin |
In , such as or , one obtains.
Out-of-process Code Component Used by COM Clients Microsoft Corporation Executable File Microsoft Corporation PDP-10 Page-Mapped Executable Binary Microsoft Programmer Settlers 4 Saved Game Ubisoft Entertainment Win32 Executable PowerBASIC PowerBASIC Inc | Proofs of all these statements may be found in the book by Isaac Chavel |
---|---|
Eigenvalues of the Laplace—Beltrami operator Lichnerowicz—Obata theorem [ ] Let M denote a compact Riemannian manifold without boundary | Like the Laplacian, the Laplace—Beltrami operator is defined as the divergence of the gradient, and is a taking functions into functions |
2002 , Riemannian Geometry and Geometric Analysis, Berlin: Springer-Verlag,.
10It is named after and | Not to be confused with |
---|---|
For any twice- real-valued function f defined on Euclidean space R n, the Laplace operator also known as the Laplacian takes f to the of its vector field, which is the sum of the n second derivatives of f with respect to each vector of an orthonormal basis for R n | One can also give an intrinsic description of the Laplace—Beltrami operator on the sphere in a |