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do Carmo M.P. — Riemannian geometry
do Carmo M.P. — Riemannian geometry

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Название: Riemannian geometry

Автор: do Carmo M.P.

Аннотация:

This text has been adopted at: University of Pennsylvania, Philadelphia University of Connecticut, Storrs Duke University, Durham, NC California Institute of Technology, Pasadena University of Washington, Seattle Swarthmore College, Swarthmore, PA University of Chicago, IL University of Michigan, Ann Arbor "In the reviewer's opinion, this is a superb book which makes learning a real pleasure." ¿ Revue Romaine de Mathematiques Pures et Appliquees "This main-stream presentation of differential geometry serves well for a course on Riemannian geometry, and it is complemented by many annotated exercises." ¿ Monatshefte F. Mathematik "This is one of the best (if even not just the best) book for those who want to get a good, smooth and quick, but yet thorough introduction to modern Riemannian geometry." ¿ Publicationes Mathematicae Contents: Differential Manifolds * Riemannian Metrics * Affine Connections; Riemannian Connections * Geodesics; Convex Neighborhoods * Curvature * Jacobi Fields * Isometric Immersions * Complete Manifolds; Hopf-Rinow and Hadamard Theorems * Spaces of Constant Curvature * Variations of Energy * The Rauch Comparison Theorem * The Morse Index Theorem * The Fundamental Group of Manifolds of Negative Curvature * The Sphere Theorem * Index Series: Mathematics: Theory and Applications


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1992

Количество страниц: 300

Добавлена в каталог: 14.09.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Action of a group      22 164
Action of a group, discontinuous      23 165
Action of a group, free      165
Action of a group, transitive      165
Axiom, countable basis      30
Axiom, Hausdorff      30
Axiom, of free mobility      167
Bianchi identities      91 106
Bundle, normal      134
Bundle, tangent      15
Bundle, tangent, metric in the      79 (Ex.)
Bundle, tangent, parametrization of the      15
Christoffel symbols      55 58
Clairaut's relation      78
Clifford torus      140 (Ex.)
Coefficient of conformality      169
Conformal metrics      160 181
Conformal metrics, connections of      181 (Ex.)
Connection, affine      50
Connection, compatible with the metric      53
Connection, Levi — Civita      55
Connection, normal      134
Connection, of a Riemannian submersion      186 (Ex.)
Connection, pseudo-Riemannian      59 (Ex.)
Connection, Riemannian      55
Connection, symmetric      54
Contractible subset      81 (Ex.)
Covariant derivative      50
Curvature constant      96
Curvature Gauss — Kronecker      129
Curvature mean      129 142
Curvature mean, curvature vector      133
Curvature normal      135
Curvature of a Riemannian submersion      187 (Ex.)
Curvature of horospheres and hyperspheres      184 (Ex.)
Curvature of the complex projective space      188 (Ex.)
Curvature of the hyperbolic space      162
Curvature of the sphere      131
Curvature principal      129
Curvature principal, of a Riemannian manifold      89
Curvature Ricci      97
Curvature scalar      97
Curvature scalar, as an integral      107 (Ex.)
Curvature sectional      94 131 162
Curvature sectional, constant sectional      96
Curvature sectional, geometric interpretation of      132
Curve, differentiable      6
Curve, divergent      153 (Ex.)
Curve, parametrized      42
Curve, piecewise differentiable      67
Cut locus      267
Diffeomorphism      10
Diffeomorphism, local      10
Differentiable structure      2
Distance      146
Divergence      83 (Ex.)
Divergence, as variation of volume      86 (Ex.)
Divergence, mean curvature as a      142 (Ex.)
Embedding      11 32
Energy      194
Equation, Codazzi      137
Equation, Gauss      130 135
Equation, Jacobi      111
Equation, Killing      82 (Ex.)
Equation, Ricci      135
Fields, Jacobi      111
Fields, Jacobi, on spaces of constant curvature      112
Fields, Jacobi, on symmetric spaces      121 (Ex.)
Fields, Killing      81 (Ex.)
Fields, Killing, singularities of a      82 (Ex.) 104 123 143
Fields, left invariant vector      40
Fields, parallel vector      52
Fields, variational      192
First fundamental form      35
Flat torus      42 46
Flat torus, as a minimal submanifold of ${S}_{3}$      140 (Ex.)
Flow of a vector field      28 63
Focal sot      232
Formula, for the first variation      195
Formula, for the second variation      198
Free homotopy class      254
Geodesic      61
Geodesic flow      63
Geodesic frame      83 (Ex.)
Geodesic ray      153 (Ex.)
Geodesic segment      61
Geodesic sphere      70
Geodesic triangle      258
Geodesic, closed      254
Geodesic, in a Riemannian manifold      153
Geodesic, in a Riemannian manifold, ray starting from      153
Geodesic, in a surface of revolution      77 (Ex.)
Geodesic, in the hyperbolic space      162
Geodesic, in the sphere      66
Geodesic, lasso      254
Geodesic, minimizing      67
Geodesic, normalized      61
Geodesic, radial      70
Geodesic, translation along a      257
Geometry, elliptic      168
Geometry, hyperbolic      168
Geometry, non-Euclidean      167
Geometry, spherical      168
Gradient      83 (Ex.)
hessian      141 (Ex.) 277
Homogeneous space      154 (Ex.)
Horosphere      178
Hyperbolic plane      46 (Ex.)
Hyperbolic plane, completeness of      153 (Ex.)
Hyperbolic plane, connection of      58 (Ex.)
Hyperbolic plane, geodesics of      73
Hypersphere      178
Hypersurface      129
Immersion      11
Immersion, applications of Index Lemma to      221
Immersion, codimension of a      11
Immersion, isometric      39
Immersion, J.D. Moore Theorem      221
Immersion, minimal      133
Immersion, totally geodesic      132
Immersion, totally geodesic, in a Lie group      140 (Ex.)
Immersion, umbilic      182
Index form      242
Index of a quadratic form      243
Index of a quadratic form, of a critical point      277
Infinitesimal isometry      see “Killing field”
Injectivity radius      271
Interior product      84 (Ex.)
inversion      169
Isometry      38
Isometry, infinitesimal      81 (Ex.)
Isometry, local      39
Jacobi identity      27
Klein bottle      25
Klein bottle, embedding in ${R}^{4}$ of      33 (Ex.)
Klein bottle, nonorientability of      33 (Ex.)
Laplacian      83 (Ex.) 143
Lemma, Index      212
Lemma, Index, for focal points      234
Lemma, Klingenberg      236 (Ex.)
Lemma, of Berger      282
Lemma, of Gauss      69
Lemma, Otsuki      225
Lie group      39
Lie group, connection of a      103 (Ex.)
Lie group, curvature of a      103 (Ex.)
Lie group, geodesies of a      80 (Ex.)
Lie group, Lie algebra of a      40
Lie group, one-parameter subgroup      80 (Ex.)
Line      252 (Ex.)
Lobatchevski plane      see “Hyperbolic plane”
Locus, conjugate      117
Locus, cut      267
Manifold, complete      145
Manifold, differentiable      2
Manifold, Einstein      108 (Ex.)
Manifold, extendible      145
Manifold, focal point free      233
Manifold, homogeneous      154 (Ex.)
Manifold, orientable      18
Manifold, oriented      18
Manifold, product      32
Manifold, Riemannian      38
Map, conformal      168
Map, differentiable      5
Map, differential of a      10
Map, exponential      65
Map, Gauss spherical      129
Map, orientation preserving      18
Metric, bi-invariant      40
Metric, bi-invariant, on a compact Lie group      46 (Ex.)
Metric, covering      152 (Ex.)
Metric, flat torus      42
Metric, induced by a covering      165
Metric, induced by an immersion      39
Metric, left invariant      40
Metric, Lorentzian      59 (Ex.)
Metric, non-euclidean of Lobachevski      46 (Ex.)
Metric, product      42
Metric, pseudo-Riemannian      58 (Ex.)
Metric, Riemannian      38 79
Mobius band      25 33
Mobius band, infinite      33 (Ex.)
Mobius band, nonorientability of      32 (Ex.)
Multiplicity of a conjugate point      116
Neighborhood, convex      76
Neighborhood, coordinate      2
Neighborhood, normal      70
Neighborhood, normal, strongly      74
Neighborhood, totally normal      72
Normal ball      70
Normal coordinates      86 (Ex.)
Nullity of a quadratic form      243
Orientable double covering      34 (Ex.)
Orientation      18
Parallel transport      52
Parametrization      2
Parametrization, associated basis to a      9
Parametrized surface      67
Partition of unity      30
Point, conjugate      116
Point, conjugate, multiplicity      116
Point, critical      17 277
Point, cut      267
Point, focal      230
Polar coordinates      122 (Ex.)
Pole      151 154
Principal directions      129
Projective plane, embedding in ${R}^{4}$ of      32 (Ex.)
Projective plane, nonorientability of      32 (Ex.)
Regular surface      16
Relation of Clairaut      78 (Ex.)
Second fundamental form      128
Space forms      164
Space, ball model      177
Space, complex projective      187 (Ex.)
Space, complex projective, curvature of      188
Space, constant curvature of      156
Space, Euclidean      39
Space, hyperbolic      160
Space, hyperbolic, completeness of the      162
Space, hyperbolic, isometries of the      175
Space, hyperbolic, models of      160 177 180
Space, lens      181
Space, real projective      4
Space, real projective, metric on      45
Space, real projective, parametrizations of      4 20
Space, symmetric (locally)      105 (Ex.)
Space, symmetric (locally), geometric characterization of      190 (Ex.)
Space, symmetric (locally), Jacobi Melds on a      121 (Ex.)
Space, tangent      9
Sphere theorem      265
Stereographic projection      19
Stereographic projection, of hyperbolic space      184 (Ex.)
Strongly convex      74
Submanifold      11 see
Submanifold, minimal      133
Submanifold, totally geodesic      132
Submanifold, umbilic      182 (Ex.)
Submanifold, umbilic, in euclidean space      184 (Ex.)
Submanifold, umbilic, in hyperbolic space      184 (Ex.)
Submersion      185 (Ex.)
Submersion, connection of a      186
Submersion, curvature of a      187
Submersion, horizontal vector of a      186 (Ex.)
Submersion, Riemannian      185 (Ex.)
Submersion, vertical vector of a      185 (Ex.)
Support of a function      30
System of coordinates      2
Tangent vector      7
Tangent vector, to a curve      6
Tensor      100
Tensor, covariant derivative of a      102
Tensor, curvature      101
Tensor, metric      101 103
Tensor, metric, on a Riemannian manifold      100
Tensor, Ricci      98
Theorem, Morse Index      243
Theorem, of Bonnet — Myers      201 208 250
Theorem, of Cartan, on closed geodesies      255
Theorem, of Cartan, on determination of the metric      157
Theorem, of E. Hopf      85 (Ex.)
Theorem, of Gauss      130
Theorem, of Hadamard      149 237
Theorem, of Hopf and Rinow      146
Theorem, of Jacobi      248
Theorem, of Liouville, on conformal transformations      170
Theorem, of Liouville, on the geodesic flow      86 (Ex.)
Theorem, of Preissman      260
Theorem, of Rauch      215
Theorem, of Rauch, on focal points      234
Theorem, of Schur      106 (Ex.)
Theorem, of Sturm      238 (Ex.) 240
Theorem, of Synge      206
Theorem, of Synge — Weinstein      203
Theorem, sphere      265
Trajectory of a vector field      28
Value, critical      17
Value, regular      17
Variation      192
Variation, proper      192
Vector field      25
Vector field, along a curve      43
Vector field, along a surface      68
Vector field, as a derivation      28
Vector field, bracket of      27
Vector field, horizontal lift of a      186 (Ex.)
Vector field, trajectory      28
Vertex of a curve      67
Volume, element      45 84 86
Volume, form      45
Volume, in a Riemannian manifold      44
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