The Laplacian on a Riemannian Manifold: An Introduction to
ISBN: , SKU: , AUTHOR: Rosenberg, Steven / Series, C. M. / Bruce, J. W., PUBLISHER: Cambridge University Press, This text on ong>anong>alysis on Riemong>anong>niong>anong> mong>anong>ifolds is a thorough introduction ong>toong> ong>toong>pics covered in advong>anong>ced research monographs on Atiyah-Singer index ong>theong>ory. ong>Theong> main ong>theong>me is ong>theong> study of heat flow associated ong>toong> ong>theong> Laplaciong>anong>s on differential forms. This provides a unified treatment of Hodge ong>theong>ory ong>anong>d ong>theong> supersymmetric proof of ong>theong> Chern-Gauss-Bonnet ong>theong>orem. In particular, ong>theong>re is a careful treatment of ong>theong> heat kernel for ong>theong> Laplaciong>anong> on functions. ong>Theong> author develops ong>theong> Atiyah-Singer index ong>theong>orem ong>anong>d its applications (without complete proofs) via ong>theong> heat equation method. Rosenberg also treats zeta functions for Laplaciong>anong>s ong>anong>d ong>anong>alytic ong>toong>rsion, ong>anong>d lays out ong>theong> recently uncovered relation between index ong>theong>ory ong>anong>d ong>anong>alytic ong>toong>rsion. ong>Theong> text is aimed at students who have had a first course in differentiable mong>anong>ifolds, ong>anong>d ong>theong> author develops ong>theong> Riemong>anong>niong>anong> geometry used from ong>theong> beginning. ong>Theong>re are over 100 exercises with hints.