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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

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In , such as or , one obtains.

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Alternatively, the operator can be generalized to operate on using the divergence and
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Examples [ ] Many examples of the Laplace—Beltrami operator can be worked out explicitly
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Flanders, Harley 1989 , Differential forms with applications to the physical sciences, Dover,• Even though some BIN files must be opened in a program for which it was developed binary format , you may still be able to open it in a universal file viewer such as File Magic
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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

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2002 , Riemannian Geometry and Geometric Analysis, Berlin: Springer-Verlag,.

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Analogous sharp bounds also hold for other Geometries and for certain degenerate Laplacians associated with these geometries like the after on a compact
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Let X i be a basis of tangent vector fields not necessarily induced by a coordinate system
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On a it is an , while on a it is
It 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

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Chanillo, Sagun, Chiu, Hung-Lin and Yang, Paul C.

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The resulting operator is called the Laplace—de Rham operator named after
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Neither the gradient nor the divergence actually depends on the choice of orientation, and so the Laplace—Beltrami operator itself does not depend on this additional structure
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The operator can be extended to operate on tensors as the divergence of the covariant derivative