Morgan F. — Riemannian geometry, a beginners guide :: Электронная библиотека попечительского совета мехмата МГУ

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Morgan F. — Riemannian geometry, a beginners guide

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Название: Riemannian geometry, a beginners guide

Автор: Morgan F.

Язык:

Рубрика: Математика/Геометрия и топология/Дифференциальная геометрия/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1993

Количество страниц: 121

Добавлена в каталог: 20.04.2005

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Предметный указатель

Arc length      18 101
Area, and Gauss curvature      22
Bending      22
Bianchi's first identity      35 41
Black hole      60
Bonnet's theorem      77 84
Calibration      99
Catenoid      15 20
Chain rule      17
Christoffel symbols ${\Gamma}^{i}_{jk}$       40 102
Co variant derivative      36 42 103
Co variant derivative, along curve      48
Comparison theorem of Rauch      88
complete      77
Conjugate points      82
Conjugate points, and sectional curvature      83
Conjugate points, and shortest geodesies      83
Connection      41
Contravariant      40
Convex norm      89 92
coordinates      14 39
Coordinates, normal      78
Costa's surface      17
Covariant      40
Crystals      89
Curvature      1 5 see “Gauss “Mean “Principal “Ricci “Riemannian “Scalar “Sectional
Curvature of curves, generalized ${D}^{2}\Phi(\kappa)$       93 97
Curvature of curves, geodesic ${\kappa}_{g}$       47
Curvature of curves, radius of      5
Curvature vector k of curve      5 47 101
CURVES      5 see “Geodesics”
Curves, curvature vector k      5
Curves, isoperimetric      93 95 97
Curves, normal vector n      5
Curves, tangent vector T      5
Cut points      83
Cylinder      22 84
Diameter      77
Diameter, Bonnet's theorem      84
Diameter, Myers theorem      87
Einstein tensor Gj      58
Ellipsoid      23
Enneper's surface      16
Equivalence principle      56 58
Euler characteristic $\chi$       67 71
Euler's formula      13
Exponential map $\rm {Exp}_{p}$       77 82
Exponential map $\rm {Exp}_{p}$ , and sectional curvature      83
Extrinsic      1
Fermat problem      98
First fundamental form      18
Gauss curvature G      1 13 19 26 27 42 101 102 103
Gauss curvature G, and area      22
Gauss curvature G, and parallel transport      72
Gauss curvature G, constant      50 87
Gauss curvature G, in local coordinates      72
Gauss curvature G, of geodesic triangle      66
Gauss curvature G, of hypersurface      71
Gauss curvature G, of projections      26
Gauss map n      69 71
Gauss — Bonnet formula      65
Gauss — Bonnet formula, proof in ${R}^{3}$       74
Gauss — Bonnet theorem      65 67—69
Gauss — Bonnet theorem, proof in $\rm {R}^{3}$       74
Gauss — Bonnet — Chern theorem      71
Gauss's Theorema Egregium      2 21 65
General relativity      55—63
Geodesic triangle      66 74
Geodesies      47 77
Geodesies, converge for positive curvature      51 58 85
Geodesies, existence and uniqueness      48
Geodesies, formula for      49 102
Geodesies, hyperbolic space      50
Geodesies, relativity      57
Geodesies, sphere      53 54
Geometric measure theory      93
Geometry, bounded      84
Geometry, global      77
Geometry, Riemannian      1 56
Gradient      103
Graph, curve      5
Graph, surface      18 19 102
Gravity      56 58
Helicoid      15 23
Helix      8
Hopf — Rinow theorem      48 79
Hyperbolic geometry      49 87

Hypersurfaces      32 33
Injectivity radius      84
INTRINSIC      2 18 21 22 39—54
Isoperimetric problem      93—98
Jacobi field      82
Laplacian      103
Length      15 18 101
Lunes      74 75
Mean curvature H      1 13 19 32 101 102
Mean curvature vector H      26 27 32 102
Mercury      55 60—62
Metric ${ds}_{2} = {g}_{ij}{du}_{i}{du}_{j}$       18 57 101
Metric ${ds}_{2} = {g}_{ij}{du}_{i}{du}_{j}$ , I to first order      21 42 78
Metric ${ds}_{2} = {g}_{ij}{du}_{i}{du}_{j}$ , Lorentz      57 60
Metric ${ds}_{2} = {g}_{ij}{du}_{i}{du}_{j}$ , Schwarzschild      58 60
Minimal surfaces      14—17
Myers theorem      87
Nash embedding theorem      39 71
networks      98—100
Non-Euclidean geometry      50
Normal coordinates      78
Normal n, to curve      5
Normal n, to hypersurface      32
Normal n, to surface      11
Norms      89—100
Norms, crystalline      98
Norms, first variation of      93
Norms, Manhattan      99
Norms, rectilinear      99
Orthogonal group SO(n)      78 79
Osculating circle      5
Parallel transport      72 74
Parallel transport, and second variation      85 86
Precession      55
Pressure      14
Principal curvatures      11 13 19 26
Principal directions      13
Projection      44
Proper time $\tau$       58
Rauch comparison theorem      88
Ricci curvature      35 41 103
Ricci curvature, Myers theorem      87
Ricci's identity      43
Ricci's lemma      43
Riemannian curvature tensor      34 41 102
Riemannian curvature tensor, and inequality of mixed partials      43
Riemannian curvature tensor, and parallel transport      72—73
Scalar curvature R      36 42 103
Scalar curvature R, and volume      42
Scherk's surface      15
Schwartz symmetrization      94
Second fundamental form II      12 18 32 34
Second fundamental tensor II      25 31
Sectional curvature      33 42 77 103
Sectional curvature, Bonnet's theorem      84
Sectional curvature, conjugate points      83
Sectional curvature, constant      50 87
Sectional curvature, diameter      84
Sectional curvature, exponential map      83
Sectional curvature, of projections      33
Sectional curvature, Rauch comparison theorem      88
Sectional curvature, sphere theorem      87
Sectional curvature, weighted average of axis curvatures      33
Slice      11
Smokestack      6—10
Soap films      99
Special relativity      55 57
Sphere      18 23 47 53 67 68 69 73 75 82 87
Sphere theorem      87
Spherical trigonometry      67
Steiner problem      98
Stokes's theorem      92
Strake      6—10
Surface of revolution      23
Surfaces, in ${R}^{3}$       11
Surfaces, in ${R}^{n}$       25
Symmetries      34 41
Tangent space ${T}_{p}S$       11
Tangent vector T to curve      5
Torsion      41
Torus      51
Variation, of curve      6 7 48 93
Variation, of surface      13 26 32
Variation, second, of curve      85 86 87
Volume, and scalar curvature      42
Weingarten map Dn      69 71
Wulff shape      94
Yamabe problem      42

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