The Laplacian on a Riemannian Manifold: An Introduction to Analysis on ManifoldsCambridge University Press, 9 janv. 1997 - 172 pages 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. |
Table des matières
III | 8 |
IV | 8 |
V | 8 |
VI | 8 |
VII | 10 |
IX | 14 |
X | 17 |
XI | 22 |
XXIV | 90 |
XXV | 92 |
XXVI | 96 |
XXVII | 101 |
XXVIII | 108 |
XXIX | 111 |
XXX | 112 |
XXXI | 116 |
XIII | 27 |
XIV | 33 |
XV | 35 |
XVI | 39 |
XVII | 46 |
XVIII | 52 |
XIX | 63 |
XXI | 67 |
XXII | 79 |
XXIII | 85 |
XXXII | 128 |
XXXIII | 134 |
XXXIV | 139 |
XXXV | 144 |
XXXVI | 153 |
XXXVIII | 154 |
XXXIX | 163 |
XL | 167 |
| 172 | |
Autres éditions - Tout afficher
The Laplacian on a Riemannian Manifold: An Introduction to Analysis on Manifolds Steven Rosenberg Aucun aperçu disponible - 1997 |
The Laplacian on a Riemannian Manifold: An Introduction to Analysis on Manifolds Steven Rosenberg Aucun aperçu disponible - 1997 |
Expressions et termes fréquents
analytic torsion Atiyah-Singer index theorem Chapter Chern-Gauss-Bonnet theorem cohomology class compact manifold compute constant coordinate chart covariant derivatives curvature tensor decomposition define definition denote dim Ker dimension dimensional dvol eigenfunctions eigenvalues endomorphism Exercise exists finite formula Gårding's inequality Gauss-Bonnet theorem Gaussian curvature geodesic geometry given H₁ heat equation heat flow heat kernel heat operator Hilbert space Hint Hodge theorem HR(M implies independent induced inner product integral invariant isomorphism isospectral k-forms Laplacian Laplacian on forms Laplacian on functions Lemma Levi-Civita connection linear matrix metric g neighborhood notation one-forms oriented orthonormal basis orthonormal frame polynomial Reidemeister torsion Rham cohomology groups Ricci Riemannian manifold Riemannian metric right hand side scalar curvature short time behavior Show smooth function Sobolev surface term theory topological uo(x vector bundles vector field zero zeta function
Fréquemment cités
Page 168 - J. Cheeger and S.-T. Yau, A lower bound for the heat kernel, Commun. Pure Appl.