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Petersen P. — Riemannian Geometry
Petersen P. — Riemannian Geometry

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Название: Riemannian Geometry

Автор: Petersen P.

Аннотация:

Intended for a one year course, this volume serves as a single source, introducing students to the important techniques and theorems, while also containing enough background on advanced topics to appeal to those students wishing to specialize in Riemannian geometry. This is one of the few works to combine both the geometric parts of Riemannian geometry and the analytic aspects of the theory, while also presenting the most up-to-date research. This book will appeal to readers with a knowledge of standard manifold theory, including such topics as tensors and Stokes theorem. Various exercises are scattered throughout the text, helping motivate readers to deepen their understanding of the subject.

Important additions to this new edition include: A completely new coordinate free formula that is easily remembered, and is, in fact, the Koszul formula in disguise; An increased number of coordinate calculations of connection and curvature; General fomulas for curvature on Lie Groups and submersions; Variational calculus has been integrated into the text, which allows for an early treatment of the Sphere theorem using a forgottten proof by Berger; Several recent results about manifolds with positive curvature.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: 2nd edition

Год издания: 2005

Количество страниц: 424

Добавлена в каталог: 22.05.2008

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Angle comparison in negative curvature      167
Arzela — Ascoli lemma      299
Axis for an isometry      167
Berger spheres      7 99 108 179 296
Berger spheres, computation of curvatures      81
Betti number estimate by Bochner      208
Betti number estimate by Gallot — Meyer      212
Betti number estimate by Gromov      358
Betti number estimate by Gromov — Gallot      275
Bianchi's first identity      33
Bianchi's second identity      33
Bochner formula      331
Bochner formula for forms      218
Bochner formula for the curvature tensor      221
Bochner technique for 1-forms      205
Bochner technique for Killing fields      190
Bochner technique in general      209
Bonnet's diameter estimate      170
Bundles of frames      151
Bundles over 2-sphere      18 91 357 373
Busemann function      285
Cartan formalism      59
Cartan's theorem      164
Center of mass      164
Cheeger's lemma      319
Cheng's maximal diameter theorem      270
Christoffel symbols      31
Clifford multiplication      213
Codazzi equation      see also Normal curvature equation
Cohomology, Cech      383
Cohomology, de Rham      202 384
Cohomology, de Rham, compactly supported      388
Cohomology, Hodge      205
Compact embedding      302
Comparison estimates for Ricci curvature      265
Comparison estimates for sectional curvature      173
Completeness      138
Completeness of closed manifolds      125
Completeness of Gromov— Hausdorff topology      296
Completeness, geodesic      118
Completeness, metric      123
Conjugate point      50 140 176
Conjugate radius estimate      176
Connectedness Lemma for functions      180
Connectedness Lemma for the energy functional      180
Connectedness Principle      181
Connectedness Principle with symmetries      199
Connection along curves      153
Connection form      59 103
Connection of bi-invariant metric      see also Metric bi-invariant
Connection on Euclidean space      23
Connection on Lie group      56
Connection on vector bundle      209
Connection, affne      26
Connection, representation in a frame      58
Connection, Riemannian      26
Constant curvature      36
Constant curvature, global classiffication      146
Constant curvature, local characterization      135
Contractions      53
Convergence of maps, Gromov — Hausdorff      298
Convergence of pointed spaces, Gromov — Hausdorff      298
Convergence of spaces, Gromov — Hausdorff      294
Convergence of spaces, Hausdorff      294
Convergence of spaces, in Hoelder topology      309
Convergence theorem of Anderson      326
Convergence theorem of Cheeger — Gromov      318
Convexity radius      177
Coordinates, Cartesian, in Euclidean space      23
Coordinates, Cartesian, on Riemannian manifold      133
Coordinates, distance      147 318
Coordinates, exponential      132
Coordinates, harmonic      304
Coordinates, normal at a point      56
Coordinates, polar, in the plane      10
Coordinates, representation of metric in      9
Covariant derivative      27
Covariant derivative in parallel frame      158
Covering space      see also Riemannian
Critical point estimate      350
Critical point theory      333
Curvature form      59 103
Curvature in dimension 3      38 60
Curvature of a tensor      35
Curvature of product metric      see also Product
Curvature of tensors      224
Curvature, constant      see also Constant curvature
Curvature, directional      36 44
Curvature, Gauss      101
Curvature, Isotropic      61 330
Curvature, operator      36 61 212 221 226 358
Curvature, operator, classiffication of $\ge 0$      260
Curvature, operator, on symmetric spaces      255
Curvature, representation in a frame      58
Curvature, Ricci      38 265 324
Curvature, Ricci, in harmonic coordinates      305
Curvature, Riemannian      33
Curvature, scalar      39 213
Curvature, sectional      36 61 153 225 333
Cut locus      142
de Rham's decomposition theorem      253
de Rham's theorem      386
Degree of a map      389
Dirac operator      213
Dirac operator on forms      215
Directional derivative      22
Dirichlet problem      304
Displacement function      168
Distance function      41 47 125 133 135
Divergence      28 57
Doubly warped products      see also Products
Eguchi — Hanson metric      91
Einstein constant      38
Einstein metric      38 254
Einstein notation      8
Einstein tensor      61
Elliptic estimates      303
Elliptic operators      303
Energy functional      126
Euclidean space      2
Euclidean space, curvature of      35
Euler characteristic      102
Exponential map      131
Extrinsic geometry      44 95
Fibration      151
Finiteness theorem for diffeomorphism types      315 320
Finiteness theorem for diffeomorphism types, in positive curvature      319
Finiteness theorem for fundamental groups      277
Finiteness theorem for homotopy types      365
Focal point      50
Frame bundle      190
Frame, left invariant      10
Frame, normal at a point      27 56
Frame, representation of metric in      10
Framing      see also Frame
Frankel's theorem      182 185
Functional distance      see also Metric
Fundamental equations for curvature      41
Fundamental equations of Riemannian geometry      47
Fundamental theorem of convergence theory      311
Fundamental theorem of hypersurface theory      100
Fundamental theorem of Riemannian geometry      25
Gauss equation      see also Tangential curvature equation
Gauss lemma      133
Gauss map      95
Gauss — Bonnet theorem      101
Geodesic      116 131 133
Gradient      22
Grassmannian, compact, as a symmetric space      244
Grassmannian, compact, computation of curvatures      246
Grassmannian, hyperbolic, as a symmetric space      246
Grassmannian, hyperbolic, computation of curvatures      248
Hadamard theorem      96
Hadamard — Cartan theorem      162
Harmonic function      281
hessian      28
Hessian comparison      342
Hinge      338
Hodge star operator      203
Hodge theorem      205
Hoelder norms      301
Holonomy      252
Holonomy classiffication      258
Holonomy of symmetric spaces      258
Homogeneous space      5
Homogeneous space, completeness of      149
Homogeneous space, k-point      184
Hopf fibration      4 6 15
Hopf problem      108 193 212 358
Hopf theorem      102
Hopf — Rinow theorem      137
Hyperbolic space      74
Hyperbolic space as left-invariant metric      80
Hyperbolic space as rotationally symmetric surface      12
Hyperbolic space, geodesics in      120
Hyperbolic space, isometry group of      76
Hyperbolic space, Minkowski space model      75
Hyperbolic space, Riemann's model      74
Hyperbolic space, Rotationally symmetric model      74
Hyperbolic space, upper half plane model      74
Hypersurface in Euclidean space      42 95
Index form      185
Index Lemma      186
Index notation      54
Injectivity radius      142
Injectivity radius estimate by Cheeger      319
Injectivity radius estimate in general      178
Injectivity radius estimate in positive curvature      178
Injectivity radius estimate, generalization of Cheeger's lemma      332
Intrinsic geometry      44 95 101
Isometric immersion      see also Riemannian
Isometry group      5 189
Isometry group of Euclidean space      5
Isometry group of Hyperbolic space      76
Isometry group of the sphere      5
Isometry, distance-preserving      147
Isometry, Riemannian      see also Riemannian
Isotropy group      5
Jacobi field along a geodesic      160
Jacobi field for a distance function      48
Killing field      23 188 242
Koszul formula      25
Kuratowski embedding      297
Laplacian connection      209
Laplacian coordiante representation      57
Laplacian estimate      284
Laplacian in harmonic coordinates      305
Laplacian on forms      204
Laplacian on functions      28
Law of Cosines      340
Left invariant frame      10
Left invariant metric      see also Metric
Length comparison      184 185
Length functional      126
Length of curve in metric space      150
Length of curve in Riemannian manifold      121
Lichnerowicz formula      213
Lie group      6
Lie group, bi-invariant metric      see also Metric
Lie group, geodesics of bi-invariant metric      151
Lie group, geodesics on      121
Line      282
Local models      331
Maximum principle      58 279
Mayer — Vietoris sequence for de Rham cohomology      384
Metric as a symmetric space      236 250
Metric ball      123
Metric distance      121
Metric on frame bundle      151
Metric on tangent bundle      151
Metric, bi-invariant      18 121 151 291
Metric, computation of curvatures      68
Metric, Einstein      38
Metric, functional      126 151
Metric, homogeneous      see also Homogeneous space
Metric, Kaehler      62 232 262 263
Metric, left-invariant      6
Metric, local representation of      9
Metric, rotationally symmetric      12
Metric, scalar flat      69
Mixed curvature equation      see also normal curvature equation
Musical isomorphisms      213
Myers' diameter estimate      171
Norm estimate, using distance functions      318 321
Norm estimate, using exponential maps      318
Norm estimate, using harmonic coordinates      324
Norm of tensors      54
Norm, $C^{m, \alpha}$, for functions      302
Norm, $C^{m, \alpha}$, for manifolds      308
Norm, harmonic, for manifolds      321
Norm, weak, for manifolds      332
Norm, weighted, for manifolds      331
Normal curvature equation      45
Normal curvature equation in Euclidean space      97
Obstructions for constant sectional curvature      147
Obstructions for negative curvature operator      108
Obstructions for negative sectional curvature      162
Obstructions for nonnegative sectional curvature      357
Obstructions for positive curvature operator      108 212
Obstructions for positive Ricci curvature      213
Obstructions for positive scalar curvature      213
Obstructions for positive sectional curvature      173
Obstructions for Ricci flatness      288
Parallel curvature      238
Parallel field along curve      156
Parallel field for a distance function      50
Parallel on manifold      28
Parallel vector field      60
Partial derivatives, first      112
Partial derivatives, second      112
Partial derivatives, third      154
partials      see also Partial derivatives
Partials derivatives and curvature      156
Pinching theorem for Ricci curvature      328
Pinching theorem for sectional curvature      329
Poincare duality      387
Poincare lemma for de Rham cohomology      385
Precompactness theorem for lower Ricci curvature bounds      300
Precompactness theorem for spaces with bounded norm      312
Precompactness theorem in Gromov — Hausdorff topology      299
Preissmann's Theorem      167
Product spheres, computations of curvatures      65
Product, cartesian      17 60
Product, doubly warped      13
Product, doubly warped, computation of curvatures      71
Product, warped      64
Projective space, complex      6 17 97
Projective space, complex, as a symmetric space      248
Projective space, complex, computation of curvatures      85 249
Projective space, complex, holonomy of      263
Projective space, quaternionic      263
Projective space, real      7
Pseudo-Riemannian manifold      4
Quarter pinching      61 346
Radial curvature equation      44
Rank      237 260
Rank of a Lie group      195
Rank, rigidity in nonpositive curvature      260
Ray      282
Riemannian covering      7 144 162
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