## The Laplacian on a Riemannian Manifold: An Introduction to Analysis on ManifoldsThis text on analysis on Riemannian manifolds is a thorough introduction to topics covered in advanced research monographs on Atiyah-Singer index theory. The main theme is the study of heat flow associated to the Laplacians on differential forms. This provides a unified treatment of Hodge theory and the supersymmetric proof of the Chern-Gauss-Bonnet theorem. In particular, there is a careful treatment of the heat kernel for the Laplacian on functions. The author develops the Atiyah-Singer index theorem and its applications (without complete proofs) via the heat equation method. Rosenberg also treats zeta functions for Laplacians and analytic torsion, and lays out the recently uncovered relation between index theory and analytic torsion. The text is aimed at students who have had a first course in differentiable manifolds, and the author develops the Riemannian geometry used from the beginning. There are over 100 exercises with hints. |

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This book is available from Steven's page.

http://math.bu.edu/people/sr/

"Feel free to print it out, but consider making a donation to a good cause in lieu of buying the text. I would like to thank Cambridge University Press for allowing me to make the text available online, in contrast to the attitude of other math text publishers."

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The Laplacian on a Riemannian Manifold: An Introduction to Analysis on Manifolds Steven Rosenberg No preview available - 1997 |

The Laplacian on a Riemannian Manifold: An Introduction to Analysis on Manifolds Steven Rosenberg No preview available - 1997 |

### Common terms and phrases

analytic torsion asymptotic Atiyah-Singer index theorem Chapter Chern-Gauss-Bonnet theorem coefficients cohomology class compact manifold compute constant coordinate chart covariant derivatives curvature tensor decomposition define definition denote differential operator dimension dvol eigenfunctions eigenvalues elliptic operators endomorphism Euler characteristic Euler form Exercise exists expression fc-forms formula Garding's inequality Gauss-Bonnet theorem Gaussian curvature geodesic geometry given gives heat equation heat flow heat kernel heat operator Hilbert space Hint Hodge theorem implies independent induced inner product integral isomorphism isospectral JM JM Laplacian Laplacian on forms Laplacian on functions Lemma Levi-Civita connection linear matrix metric g neighborhood notation one-forms oriented orthonormal basis orthonormal frame parallel translation pointwise polynomial Reidemeister torsion result Rham cohomology groups Riemannian manifold Riemannian metric right hand side scalar curvature short time behavior Show signature smooth function spectrum surface term theory vanishes vector bundles vector field zero zeta function

### Popular passages

Page 168 - J. Cheeger and S.-T. Yau, A lower bound for the heat kernel, Commun. Pure Appl.