Rosenberg S. — The Laplacian on a Riemannian manifold :: Электронная библиотека попечительского совета мехмата МГУ

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Rosenberg S. — The Laplacian on a Riemannian manifold

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Название: The Laplacian on a Riemannian manifold

Автор: Rosenberg S.

Аннотация:

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.

Язык:

Рубрика: Математика/Геометрия и топология/Дифференциальная геометрия/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1997

Количество страниц: 180

Добавлена в каталог: 20.04.2005

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Предметный указатель

$H_{s}\Lambda^{k}M$       34
$L^{2}(M,g)$       16
$L^{2}\Lambda^{k}T^{\bullet}M$       17
$T^{l}_{k}$       60
Acyclic representation      152
Analytic torsion      153
Asymptotic expansion      99
Atiyah — Singer index theorem      141
Basic elliptic estimate      36
Berezin integral      125
Betti number      42
Bianchi identity      60 127
Bismut — Lott theorem      163
Bochner's formula      75
Bochner's theorem      75
Cartan — Hadamard theorem      62
Characteristic class      129
Characteristic form      129
Chern character      138
Chern classes      132
Chern — Gauss — Bonnet theorem      112
Chern — Weil theory      126
Christoffel symbols      55
Compactness Theorem      25
Complete polarization      127
Conformal metric      16
Connected sum      135
Connection on a vector bundle      131
Connection, flat      153
Connection, Levi-Civita      64 65
Connection, one-forms      126
Contraction of a tensor      61
Convolution of functions      23
Covariant derivative      64
Critical path      78
Curl      22
Curvature of a general connection      131
Curvature, endomorphism      70
Curvature, Gaussian, for arbitrary surfaces      55
Curvature, Gaussian, for surfaces in $R^{3}$       53
Curvature, Ricci      60 74
Curvature, scalar      61
Curvature, sectional      62
Curvature, two-forms      126
de Rham cohomology groups      40 42
de Rham's theorem      45
Divergence      18
Donaldson theory      133
Double form      34
Duhamel's formula      94
Einstein summation convention      14
Elliptic operator      139
Elliptic operator, determinant of      153
Elliptic operator, symbol matrix of      139
Euclidean space      11
Euler characteristic      49
Euler class      131
Euler form      112
Exponential map      80
Families index theorem      162
Fefferman — Graham invariant      148
Fermion calculus      67
Feynman — Kac formula      125
Fourier transform      22
Garding's inequality      36 75
Gauss map      53
Gauss — Bonnet theorem      56 111
Gauss' lemma      82
Geodesic      79
Geodesic, minimal      80
Global inner product      16
Gradient      17
Green's formula      156
Harmonic forms      46
Heat equation      5 27
Heat kernel for forms      99
Heat kernel for functions      98
Heat kernel on a manifold      27
Heat kernel on Euclidean space      6
Heat kernel on forms      34
Heat kernel on the circle      6
Heat kernel on the real line      6
Heat kernel, approximate on forms      114
Heat kernel, approximate on functions      114
Heat kernel, asymptotic expansion      99
Heat kernel, dependence on metric      155
Heat kernel, pointwise convergence      89
Heat kernel, positivity      106
Heat kernel, symmetry      8
Heat operator      27
Heat operator, semigroup property      28
Higher torsion classes      164
Higher torsion forms      164
Hirzebruch L-polynomial      135

Hirzebruch proportionality      113
Hirzebruch signature theorem      136
Hirzebruch — Riemann — Roch theorem      139
Hodge decomposition theorem      38
Hodge inner product      21
Hodge star operator      19
Hodge theorem      46
Hodge theorem for forms      34 37
Hodge theorem for functions      32
Holonomy group      67
Homotopy equivalence      42
Hopf conjectures      62 113
Hopf — Rinow theorem      80
Horizontal distribution      162
Index bundle      161
Index o? an operator      133
Integrability condition      40 54 57
Integration along the fibers      162
Interior product      67
Intersection pairing      132
Invariant polynomial      127
Isospectral manifolds      100
Jacobi equation      85
Jacobi fields      85
Jacobi's formula      9
Join      51
Kronecker delta      15
Laplacian in polar coordinates      87
Laplacian of an exact complex      152
Laplacian on forms      33
Laplacian on functions      18 19 21
Laplacian on twisted forms      153
Laplacian, Bochner      72 125
Laplacian, conformal      145
Laplacian-type operator      142
Lie bracket      126
Local frame      126
Lohkamp's theorem      61
McKean — Singer lemma      110
Mellin transform      143
Myers' Theorem      61 75
Operator, compact      2
Operator, elliptic      139
Operator, Lapladan-type      142
Orthonormal frame      70
Parallel translation      66 67
Parametrix      93
Piaffian      129 130
Pontrjagin classes      129
R-torsion      see "Analytic torsion"
Rayleigh — Ritz formula      159
Reidemeister torsion      152
Riemami's theorem      57
Riemann curvature tensor      57 67
Riemannian distance      13
Riemannian manifold      11
Riemannian manifold, isometry of      11
Riemannian metric      11
Riemannian metric, flat      57
Riemannian metric, induced      11
Riemannian normal coordinates      82
Riemannian polar coordinate      82
Schroedinger operator      125
Signature      133
Signature operator      133
Signature operator, twisted      138 161
Sobolev embedding theorem      24
Sobolev spaces for forms      33
Sobolev spaces for functions on Euclidean space      22
Sobolev spaces for Junctions on manifolds      26
Spectral theorem      2
Spectrum      2 4
Sunada's theorem      149
Supertrace      69
Symbol bundle      141
Symbol matrix      139
Synchronous frame      73
Synge's theorem      62
Tangent bundle along the fibers      162
Theorema Egreglum      55
Todd polynomial      138
Torsion of an exact complex      152
Traceless Ricci tensor      148
Transport equations      92
Universal polynomials      83
Variation vector field      79
Virtual bundle      140 101
Volume      15
Volume form      15
Weierstrass approximation theorem      7
Weitzenboeck formula      72
Weyl tenBor      148
Wiener measure      126
Zeta function      143 153
Zeta function, meromorphic continuation      143

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