Abstract
In this chapter we will generalize the Laplacian on Euclidean space to an operator on differential forms on a Riemannian manifold. By a Riemannian manifold, we roughly mean a manifold equipped with a method for measuring lengths of tangent vectors, and hence of curves. Throughout this text, we will concentrate on studying the heat flow associated to these Laplacians. The main result of this chapter, the Hodge theorem, states that the long time behavior of the heat flow is controlled by the topology of the manifold.
Keywords
Pseudo-Riemannian manifoldLaplace operatorRiemannian manifoldManifold (fluid mechanics)Hermitian manifoldExponential map (Riemannian geometry)MathematicsClosed manifoldStatistical manifoldTangent spaceEuclidean spacePure mathematicsInvariant manifoldSpace (punctuation)Differential geometryTopology (electrical circuits)Laplace–Beltrami operatorMathematical analysisRicci curvatureInformation geometryComputer sciencep-LaplacianScalar curvatureCombinatoricsGeometrySectional curvature
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Publication Info
- Year
- 1997
- Type
- book-chapter
- Pages
- 1-51
- Citations
- 367
- Access
- Closed
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Cite This
Steven Rosenberg
(1997).
The Laplacian on a Riemannian Manifold.
Cambridge University Press eBooks
, 1-51.
https://doi.org/10.1017/cbo9780511623783.002
Identifiers
- DOI
- 10.1017/cbo9780511623783.002