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Lee J.M. — Riemannian Manifolds: an Introduction to Curvature
Lee J.M. — Riemannian Manifolds: an Introduction to Curvature



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Название: Riemannian Manifolds: an Introduction to Curvature

Автор: Lee J.M.

Аннотация:

This text is designed for a one-quarter or one-semester graduate course on Riemannian geometry. It focuses on developing an intimate acquaintance with the geometric meaning of curvature and thereby introduces and demonstrates all the main technical tools needed for a more advanced study of Riemannian manifolds. The book begins with a careful treatment of the machinery of metrics, connections, and geodesics, and then introduces the curvature tensor as a way of measuring whether a Riemannian manifold is locally equivalent to Euclidean space. Submanifold theory is developed next in order to give the curvature tensor a concrete quantitative interpretation. The remainder of the text is devoted to proving the four most fundamental theorems relating curvature and topology: the Gauss-Bonnet Theorem, the Cartan-Hadamard Theorem, Bonnet's Theorem, and the characterization of manifolds of constant curvature. This unique volume will appeal especially to students by presenting a selective introduction to the main ideas of the subject in an easily accessible way. The material is ideal for a single course, but broad enough to provide students with a firm foundation from which to pursue research or develop applications in Riemannian geometry and other fields that use its tools.


Язык: en

Рубрика: Математика/Геометрия и топология/Дифференциальная геометрия/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1997

Количество страниц: 224

Добавлена в каталог: 19.04.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
# (sharp)      28—29
$B_R(p)$ (geodesic ball)      106
$d(p, q)$ (Riemannian distance)      94
$dV$ (Riemannian volume element)      29
$dV_g$ (Riemannian volume element)      29
$D_s$ (covariant derivative along transverse curves)      97
$D_t$ (covariant derivative along a curve)      57
$g^\circ$ (round metric)      33
$g^\circ_R$ (round metric of radius R)      33
$h_R$ (hyperbolic metric)      38—41
$i_X$ (interior multiplication)      43
$L(\gamma)$ (length of curve)      92
$L_g(\gamma)$ (length of curve)      92
$O_+(n,1)$ (Lorentz group)      41
$P_{t_0t_1}$ (parallel translation operator)      61
$r(x)$ (radial distance function)      77
$R_C$ (Ricci tensor)      124
$R_m$ (curvature tensor)      118
$S_R(p)$ (geodesic sphere)      106
$tr_g$ (trace with respect to g)      28
$T\tilde M|\phantom{}_M$ (ambient tangent bundle)      132
$T^*M$ (cotangent bundle)      17
$T^k(V)$ (space of covariant k-tensors)      12
$T^k_l(V)$ (space of mixed tensors)      12
$T^k_lM$ (bundle of mixed tensors)      19
$T_l(V)$ (space of contravariant l-tensors)      12
$\chi(M)$ (Euler characteristic)      167
$\Delta$ (Laplacian)      44
$\dot\gamma$ (velocity vector)      56
$\dot\gamma(a_i^\pm)$ (one-sided velocity vectors)      92
$\flat$ (flat)      27—29
$\Gamma(s, t)$ (admissible family)      96
$\gamma_V$ (geodesic with initial velocity V)      59
$\kappa_N(t)$ (signed curvature)      163
$\Lambda^kM$ (bundle of k-forms)      20
$\mathbf B_R^n$ (Poincare ball)      38
$\mathbf H^n_R$ (hyperbolic space)      38—41
$\mathbf R^n$ (Euclidean space)      25 33
$\mathbf U^n_R$ (Poincare half-space)      38
$\mathbf{S}^n$ (unit n-sphere)      33
$\mathbf{S}^n_R$ (n-sphere of radius R)      33
$\mathcal N(M)$ (space of sections of normal bundle)      133
$\mathcal T(M)$ (space of vector fields)      19
$\mathcal T(\gamma)$ (space of vector fields along a curve)      56
$\mathcal T(\tilde M|\phantom{}_M)$ (space of sections of ambient tangent bundle)      133
$\mathcal T^1(M)$ (space of 1-forms)      20
$\mathcal T^k(M)$ (space of covariant tensor fields)      20
$\mathcal T^k_l(M)$ (space of mixed tensor fields)      20
$\mathrm{exp}_p$ (restricted exponential map)      72
$\mathrm{Rot}(\gamma)$ (rotation angle)      156
$\nabla F$ (total covariant derivative)      54
$\nabla^2u$ (covariant Hessian)      54
$\nabla^\top$ (tangential connection)      66 135
$\nabla_XY$ (covariant derivative)      49—50
$\omega_i^j$ (connection 1-forms)      64
$\overline{B}_R(p)$ (closed geodesic ball)      106
$\overline{g}$ (Euclidean metric)      25
$\partial/\partial r$ (unit radial vector field)      77
$\partial/\partial x_i$ (coordinate vector field)      15
$\partial_i$ (coordinate vector field)      15
$\pi^\perp$ (normal projection)      133
$\pi^\top$ (tangential projection)      133
$\tau$ (torsion tensor)      63 68
$\varepsilon$ (domain of the exponential map)      72
Acceleration, Euclidean      48
Acceleration, of a curve on a manifold      58
Acceleration, of a plane curve      3
Acceleration, tangential      48
Adapted orthonormal frame      43 133
Adjoint representation      46
Admissible curve      92
Admissible family      96
Affine connection      51
Aims at a point      109
Algebraic Bianchi identity      122
Alternating tensors      14
Ambient manifold      132
Ambient tangent bundle      132
Ambrose, Cartan — Ambrose — Hicks theorem      205
Angle between vectors      23
Angle tangent      156 157
Angle-sum theorem      2 162 166
Arc length function      93
Arc length parametrization      93
Aspherical      199
Automorphism, inner      46
Ball, geodesic      76 106
Ball, Poincare      38
Base of a vector bundle      16
Berger metrics      151
Berger, Marcel      203
Bi-invariant metric      46 89
Bi-invariant metric, curvature of      129 153
Bi-invariant metric, existence of      46
Bi-invariant metric, exponential map      89
Bianchi identity, algebraic      122
Bianchi identity, contracted      124
Bianchi identity, differential      123
Bianchi identity, first      122
Bianchi identity, second      123
Bonnet, Bonnet's theorem      9 200
Bonnet, Gauss — Bonnet theorem      167
Boundary problem, two-point      184
Bundle, cotangent      17
Bundle, normal      17 133
Bundle, of k-forms      20
Bundle, of tensors      19
Bundle, tangent      17
Bundle, vector      16
Calculus of variations      96
Caratheodory metric      32
Carnot — Caratheodory metric      31
Cartan — Ambrose — Hicks theorem      205
Cartan — Hadamard manifold      199
Cartan — Hadamard theorem      9 196
Cartan's first structure equation      64
Cartan's second structure equation      128
Catenoid      150
Cayley transform      40
Cayley transform, generalized      40
Chern — Gauss — Bonnet theorem      170
Christoffel symbols      51
Christoffel symbols, formula in coordinates      70
Circle classification theorem      2
circles      2
Circumference theorem      2 162 166
Classification Theorem      2
Classification theorem, circle      2
Classification theorem, constant curvature metrics      9 206
Classification theorem, plane curve      4
Closed curve      156
Closed geodesic ball      76
Coframe      20
Commuting vector fields, normal form      121
Comparison theorem, conjugate point      195
Comparison theorem, Jacobi field      194
Comparison theorem, metric      196
Comparison theorem, Rauch      203 204
Comparison theorem, Sturm      194
Compatibility with a metric      67
Complete, geodesically      108
Complex projective space      46
Conformal metrics      35
Conformally equivalent      35
Conformally flat, hyperbolic space      41
Conformally flat, locally      37
Conformally flat, sphere      37
Congruent      2
conjugate      182
Conjugate locus      190
Conjugate point      182
Conjugate point, comparison theorem      195
Conjugate point, geodesic not minimizing past      188
Conjugate point, singularity of expp      182
Connection      49
Connection 1-forms      166
Connection, 1-forms      64 165
Connection, Euclidean      52
Connection, existence of      52
Connection, formula in arbitrary frame      69
Connection, formula in coordinates      70
Connection, in a vector bundle      49
Connection, in components      51
Connection, linear      51
Connection, naturality      70
Connection, on tensor bundles      53—54
Connection, Riemannian      68
Connection, tangential      66
Constant Gaussian curvature      7
Constant sectional curvature      148
Constant sectional curvature, classification      9 206
Constant sectional curvature, formula for curvature tensor      148
Constant sectional curvature, formula for metric      179
Constant sectional curvature, local uniqueness      181
Constant sectional curvature, model spaces      9
Constant sectional curvature, uniqueness      204
Constant speed curve      70
Contracted Bianchi identity      124
Contraction      13
Contravariant tensor      12
Control theory      32
Converge to infinity      113
Convex geodesic polygon      171
Convex set      112
coordinates      14
Coordinates, have upper indices      15
Coordinates, local      14
Coordinates, normal      77
Coordinates, Riemannian normal      77
Coordinates, slice      15
Coordinates, standard, on $\mathbf R_n$      25
Coordinates, standard, on tangent bundle      19
Cosmological constant      126
Cotangent bundle      17
Covariant derivative      50
Covariant derivative, along a curve      57—58
Covariant derivative, of tensor field      53—54
Covariant derivative, total      54
Covariant Hessian      54 63
Covariant tensor      12
Covectors      11
Covering map      197
Covering metric      27
Covering Riemannian      27
Covering transformation      27
Critical point      101 126 142
Crystallographic groups      206
Curvature      3—10 117
Curvature, 2-forms      128
Curvature, constant sectional      9 148 179—181 204 206
Curvature, constant, formula for      148
Curvature, endomorphism      117 128
Curvature, Gaussian      6—7 142—145
Curvature, geodesic      137
Curvature, in coordinates      128
Curvature, mean      142
Curvature, of a curve in a manifold      137
Curvature, of a plane curve      3
Curvature, principal      4 141
Curvature, Ricci      124
Curvature, Riemann      117 118
Curvature, scalar      124
Curvature, sectional      9 146
Curvature, signed      4 163
Curvature, tensor      118
Curve      55
Curve, admissible      92
Curve, in a manifold      55
Curve, plane      3
Curve, segment      55
Curved polygon      157 162
Cusp      157
Cut locus      190
Cut point      190
Cylinder, principal curvatures      5
Deck transformation      27
Defining function      150
Diameter      199
Difference tensor      63
Differential Bianchi identity      123
Differential forms      20
Dihedral groups      206
Distance, Riemannian      94
Divergence      43
Divergence, in terms of covariant derivatives      88
Divergence, operator      43
Divergence, Theorem      43
Domain of the exponential map      72
Dual basis      13
Dual coframe      20
Dual space      11
E(n) (Euclidean group)      44
Edges of a curved polygon      157
Eigenfunction of the Laplacian      44
Eigenvalue of the Laplacian      44
Einstein field equation      126
Einstein general theory of relativity      31 126
Einstein metric      125 202
Einstein special theory of relativity      31
Einstein summation convention      13
Embedded submanifold      15
Embedding      15
Embedding, isometric      132
End(V) (space of endomorphisms)      12
Endomorphism curvature      117
Endomorphism of a vector space      12
Escape lemma      60
Euclidean acceleration      48
Euclidean connection      52
Euclidean geodesics      81
Euclidean group      44
Euclidean homogeneous and isotropic      45
Euclidean metric      25 33
Euclidean triangle      2
Euler characteristic      167 170
Euler — Lagrange equation      101
Existence and uniqueness for linear ODEs      60
Existence and uniqueness for ODEs      58
Existence and uniqueness of geodesics      58
Existence and uniqueness of Jacobi fields      176
exp (exponential map)      72
Exponential map      72
Exponential map, domain of      72
Exponential map, naturality      75
Exponential map, of bi-invariant metric      89
Extendible vector fields      56
Extension of functions      15
Extension of vector fields      16 132
Exterior angle      157 163
Exterior k-form      14
Family, admissible      96
Fiber metric      29
Fiber of a submersion      45
Fiber of a vector bundle      16
Finsler metric      32
First Bianchi identity      122
First fundamental form      134
First structure equation      64
First variation      99
Fixed-endpoint variation      98
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