The Laplacian on a Riemannian Manifold: An Introduction to Analysis on Manifolds
This 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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assume basis bounded bundle called Chapter choice closed compact manifold complex compute Conclude condition consider constant construct coordinates curvature define definition denote depends derivatives determine differential dimension dimensional dvol eigenvalues equals equation equivalent Euler example Exercise exists expression extends exterior derivative fact finite fixed follows forms formula frame functions geodesic geometry given gives heat equation heat flow heat kernel implies independent index theorem induced inner product integral invariant isomorphism k-forms Laplacian Lemma length manifold matrix Moreover neighborhood notation Note operator oriented orthonormal particular polynomial positive proof prove reader respect result Rham cohomology Riemannian manifold Riemannian metric right hand side satisfies Show smooth solution space standard surface taking tangent tangent vectors tensor term theorem theory topological vanishes vector field write zero zeta function
Page 168 - J. Cheeger and S.-T. Yau, A lower bound for the heat kernel, Commun. Pure Appl.