The Laplacian on a Riemannian Manifold: An Introduction to Analysis on Manifolds

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Cambridge University Press, Jan 9, 1997 - Mathematics - 172 pages
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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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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." 

Contents

III
1
IV
2
V
3
VI
5
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
XLI
172
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Page 168 - J. Cheeger and S.-T. Yau, A lower bound for the heat kernel, Commun. Pure Appl.

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