Fomenko А.Т., Mishehenko A.S. — A Short Course in Differential Geometry and Topology :: Электронная библиотека попечительского совета мехмата МГУ

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Fomenko А.Т., Mishehenko A.S. — A Short Course in Differential Geometry and Topology

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Название: A Short Course in Differential Geometry and Topology

Авторы: Fomenko А.Т., Mishehenko A.S.

Аннотация:

A Short Course on Differential Geometry and Topology by Professor A.T. Fomenko and Professor A.S. Mishehenko is based on the course taught at the Faculty of Mechanics and Mathematics of Moscow State University. It is intended for students of mathematics, mechanics and physics and also provides a useful reference text for postgraduates and researchers specialising in modern geometry and its applications.

Язык:

Рубрика: Математика/

Серия: Сделано в холле

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2009

Количество страниц: 273

Добавлена в каталог: 07.11.2009

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Предметный указатель

Absolute      30
Action trajectory functional      247
Adherent point      40 41
Adjoint flow      146
Adjoint potential      146
Affine connection      193
Alternation      182
Angle between two intersecting smooth trajectories      11
Arcwise connected space      46
Arcwise connectedness      46
Atlas of charts      59
Autonomous or stationary system      141
Beltrami surface      115
Beltrami “funnel”      117
Binormal vector to the curve      101
Boundary point      150
Branch of the algebraic function      166
Branch points of the algebraic function      166
Cartesian coordinates      1
Cartesian product of the sets      43
Cartesian product of topological spaces      44
Catenoid      119
Cauchy — Riemann equation      146
Change of coordinates      5
Chart      59
Christoffel symbol      192
Closed differential form      225
Closed subset      40 41
Closure      40 41
Compact metric space      51
Compact symplectic group      138
Compact topological space      49
Compactification      167
Complex potential of the flow      147
Complex-analytic function      146
Complex-analytic manifold      64
Configuration space      80
Conformal Riemannian metric      22
Conformal surface      120
Conformal transformation      23
Conformally Euclidean metric      104
Connected component of the identity of the group      131
Connected space      46
Connectedness      45
Continuous coordinate system      3
Continuous mapping      42
Continuously differentiable function      62
Contraction operation      181
Contravariant index      175
coordinates      3
Coordinates of the tangent vector      74 75
Covariant differentiation      192
Covariant differentiation along a curve      199
Covariant index      175
Covector      174
Critical function for the functional      244
Critical point of a function      111 140
Current lines of fluid      141
Curvature in the two dimensional direction      213 216
Curvature of the curve      94 101
Curvature radius      94
Curvilinear coordinate systems      1
Curvilinear coordinates      4
Cylindrical coordinates      2 8
De Rham cohomology group      225
Deformation tensor      178
Degree of the mapping      237
Dense subset      42
Derivative of the function f in direction of the tangent vector      77
Diffeomorphism of class $C^{r}$       65
Differential of the mapping in a point      81
Differentiation of the function f in direction of the tangent vector      77
Dimension of the manifold      59
Discrete topology      41
Distance between the points      40
Distance between two sets      40
Divergence of the flow      144
Domain      3
Elliptic geometry      28
Embedding      84
Equation of parallel translation along the curve      200
Equations of geodesic      201
Equivalent atlases of charts      63
Euclidean coordinates      195
Euclidean inner product      1
Euclidean Riemannian metric      17
Euler equations for the functional      247
Euler formula      110
Exact differential form      225
Exterior differential form      184
Exterior product      183
Extremal function for the functional      244
Factor      44
Field parallel along the curve with respect to the connection      199
Finer covering      48
First quadratic form of a hypersurface      103
Frenet formulae      96 98
Frenet frame      94
Functional      241
Functions of change of coordinates      62
Fundamental domain      153
Gauss — Bonnet theorem      240
Gauss — Ostrogradskii formula      234
Gaussian curvature of the surface      107
Genus of the surface      163
Geodesic curve      201
Gradient of differential form      220
Green formula      233
Hamiltonian      257
Hamiltonian equations      259
Hamiltonian field      260
Harmonic function      146
Harmonic surface      120
Hausdorff space      47
Homeomorphic topological spaces      43
Homeomorphism      3 43
Homotopic mappings      229
Homotopy      228
Hyperbolic geometry      30
Imaginary      137
Immersion      84
Impulse      253
Incompressible flow      144 262
Indefinite metric      18
Induced Riemannian metric      21
Induced topology      41
Inertia momentum tensor      178
Integral curve of the field      141
Integral of the system      141
Interior      41
Interior point      41
Irrotational flow      145
Isometry      121 122
Isometry group      122
Isotropic cone      26
Isotropic vector      18
Jacobi matrix      4
Jacobian      4
Kepler laws      2
Klein bottle      153

Laplace operator      146
Leaf of the Riemann surface      166
Length of a curve      11
Length of a smooth curve in pseudo-Euclidean space      19
Length of vector      10
Length trajectory functional      247
Level surfaces      68
Lie algebra      265
Lie group      131
Light vector      18
Linear independent vector fields      142
Linear-fractional transformation      33
Lobachevskii geometry      30
Lobachevskii metric      32 117
Local coordinate system      6
Lowering of indices      181
Lowering of superscript      181
Loxodromic curve      23
Main theorem of algebra      238
Manifold      59
Manifold of constant curvature      216
Manifold of level с of the function f      68
Manifold of negative curvature      216
Manifold of positive curvature      216
Manifold of zero curvature      216
Manifold with boundary      150
Mean curvature of the surface      107
Metric space      39 40
Metric tensor      178
Meusnier theorem      109
Minimal geodesic      249
Minimal surface      120
Minkowski space      18
Moebius band      154
Motion      121
n-dimensional manifold      59
N-dimensional sphere      61
Natural parameter of the curve      93
Natural parameterization of the curve      93
Neighborhood of a point      41
Neighborhood of a set      41
Newton — Leibnitz formula      235
Nondegenerate linear-fractional transformation      35
Nondegenerate surface      72
Nonsingular (noncritical or regular) point      140
Nonstationary flow      141
Normal section      109
Normal topological space      52
Normal vector to the curve      94 101
Open covering      48
Open subset      40
Orientable manifold      151
Oriented atlas of charts      151
Orthogonal group      133
Parallel vector      200
Partition of unity      54
Permutation of indices      180
Phase space      80
Plane section of the hypersurface      109
Poincare lemma      231
Poincare model      30 118
Points of topological space      41
Poisson bracket      264
Polar coordinate system      1 7
Potentional or gradient field      143
Principal directions of the hypersurface      108
Principal normal section      110
Principal normal to the curve      101
Pseudosphere      25 115
Pullback of the form      227
Quaternion      137
Quaternion algebra      137
Quotient space      44
Quotient topology      44
Raising of indices      181
Raising of subscript      181
Real symplectic group      137
Real-analytic manifold      64
Realification      135
Regular coordinate system      4
Regular point of the mapping      84
Ricci tensor      213
Riemann surface for the algebraic function      164
Riemannian connection      195
Riemannian curvature tensor of the connection      210
Riemannian metric      15 87
Right helicoid      119
Second quadratic form of a hypersurface      104 106
Set of zero measure      231
Singular point for a vector field      140
Skew-symmetric field      182
Skew-symmetric gradient      259
Smooth curve      10
Smooth homeomorphism of class $C^{r}$       65
Smooth homotopv      229
Smooth manifold of class $C^{r}$       64
Smooth mapping of class $C^{r}$       65
Smooth n-dimensional manifold      62
Smooth tensor field      175
Space-like vector      18
Special orthogonal group      133
Spherical coordinates      2 8
Spherical mapping      239
Spherical neighborhood of radius $\varepsilon$       40
Stationary function for the functional      244
Stereographic projection      21
Stokes formula      232
Stress tensor      179
Submanifold      84
subspace      41
Support of the function      54
Surface of constant curvature      114
Surface of positive, zero, and negative curvature      114
Symmetric connection      193
Symmetric field      182
Symmetrization      182
Symplectic coordinates      260
Symplectic space      137
Symplectic transformation      137
Tangent bundle of the manifold M      79
Tangent space of the manifold M at the point $P_{0}$       75
Tangent vector      11 73
Tangent vector $\xi$ at the point $P_{0}$ to the manifold M      74
Tangent vector of the curve      75
Tensor law of coordinate change      75
Tensor product      181
Time-like vector      18
Topological space      41
topology      40
Torsion of the curve      101
Torsion tensor of the affine connection      193
Torus      72
Transfer functions      62
Transformation group      121
Uniform convergence      52
Uniform limit      43
Velocity vector      11
Velocity vector of the curve      75
Volume of the domain      186
World line      19

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