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Bibliografická citace

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0 (hodnocen0 x )
BK
First edition
Olomouc : Palacký University, 2008
220 stran ; 25 cm
Externí odkaz    Obsah 

objednat
ISBN 978-80-244-2168-1 (brožováno)
Monographs
Nad názvem: Palacký University, Olomouc, Faculty of Science
Obsahuje bibliografii na stranách 185-210, bibliografické odkazy a rejstříky
* geodetická zobrazení * variety s afinní konexí
001419657
CONTENTS // INTRODUCTION ? // 1 PRELIMINARIES 13 // 1.1 Curves and surfaces in Euclidean spaces ... 13 // 1.1.1 Coordinates... 13 // 1.1.2 Vector functions, continuity and differentiability . 14 // 1.1.3 Curves and parametrized curves ... 15 // 1.1.4 Frenet frame and Frenet.Serret formulas of a curve in E3 17 // 1.1.5 Surface and simple surface (patch) in E3... 19 // 1.1.6 Isometry and Inner geometry... 21 // 1.1.7 Normal and geodesic curvatures in surfaces... 21 // 1.2 Geodesics in surfaces... 24 // 1.2.1 Characterization of geodesics... 24 // 1.2.2 Existence and uniqueness of geodesics... 25 // 1.2.3 Motivations and applications... 27 // 2 MANIFOLDS WITH AFFINE CONNECTION // AND RIEMANNIAN MANIFOLDS 29 // 2.1 Manifold... 29 // 2.1.1 Topological manifold, chart... 29 // 2.1.2 Differentiable structure (complete atlas)... 30 // 2.1.3 Smooth map, diffcomorphism... 31 // 2.1.4 Tangent vector, tangent space, tangent bundle ... 32 // 2.1.5 Differential map ... 33 // 2.1.6 Curve, tangent vector of a curve ... 34 // 2.1.7 Vector field, flow... 35 // 2.1.8 One-form... 37 // 2.2 Tensor fields and geometric objects... 37 // 2.2.1 Tensors on a vector space... 37 // 2.2.2 Tensors on manifolds... 39 // 2.2.3 Geometric objects on manifolds... 41 // 2.3 Manifolds with affine connection... 43 // 2.3.1 Affine connections, manifolds with affine connection ... 43 // 2.3.2 Covariant differentiation... 44 // 2.3.3 Curvature and Ricci tensor ... 45 // 2.3.4 Flat, Ricci flat and equiaffine manifolds...
46 // 2.4 Riemannian manifolds ... 46 // 2.4.1 Riemannian metric... 46 // 2.4.2 Isometric diffeomorphisms... 48 // 2.4.3 Levi-Civita connection and Riemannian tensor... 48 // 2.4.4 Spaces of constant curvature and Einstein spaces... 50 // 7 // 8 // 2.5 On systems of partial differential equations of Cauchy type ... 51 // 2.5.1 Systems of PDEs of Cauchy type in R" ... 51 // 2.5.2 On mixed systems of PDEs of Cauchy type in R"... 52 // 2.5.3 On a mixed linear system of PDEs of Cauchy type in R" 53 // 2.5.4 Mixed PDEs in a tensor form... 53 // 2.5.5 On systems of PDEs of Cauchy type in manifolds... 54 // 2.5.6 Application... 55 // 3 VARIATIONAL PROPERTIES OF GEODESICS 57 // 3.1 Motivation and historical remarks... 57 // 3.2 Variational problem... 58 // 3.3 Variational problem of geodesics in Riemannian spaces... 59 // 3.4 Generalized variational problem for geodesics ... 61 // 3.5 Geodesics on affine manifolds ... 62 // 3.6 Some remarks on definitions for geodesics... 63 // 3.7 Applications... 65 // 4 GEODESIC MAPPINGS // OF MANIFOLDS WITH AFFINE CONNECTION 67 // 4.1 Geodesic-preserving mappings... 67 // 4.1.1 Introduction to geodesic mappings theory... 67 // 4.1.2 Examples of geodesic mappings... 69 // 4.2 Formalisms used by diffeomorphisms of manifolds... 69 // 4.2.1 Formalism of a “common coordinate system”... 70 // 4.2.2 Formalism of a “common manifold” ... 70 // 4.2.3 Deformation tensor of a mapping... 70 // 4.3 Levi-Civita equations of geodesic mappings ... 71 // 4.4 Equivalence
classes of geodesic mappings... 73 // 4.5 Some geometric objects under geodesic mappings... 74 // 4.5.1 Thomas’ projective parameter . ... 74 // 4.5.2 Riemannian and Ricci tensor under geodesic mappings . . 75 // 4.5.3 Weyl tensor of projective curvature... 75 // 4.6 Geodesic mappings of equiaffine manifolds... 77 // 4.6.1 Geodesic mappings between equiaffine manifolds... 77 // 4.6.2 Geodesic mappings onto manifolds with affine connection 78 // 4.7 Projectively flat manifolds... 80 // 4.7.1 Geodesic mappings of projectively flat manifolds... 80 // 4.7.2 Characterization of projectively flat manifolds... 81 // 5 GEODESIC MAPPINGS // ONTO RIEMANNIAN MANIFOLDS 85 // 5.1 Levi-Civita equation of GM onto Riemannian manifolds ... 85 // 5.1.1 On equations of mappings onto Riemannian manifolds . . 85 // 5.1.2 Levi-Civita equations of geodesic mappings... 86 // 5.1.3 Mikeš-Berezovski equations of geodesic mappings... 87 // 5.1.4 On mobility degree with respect to geodesic mappings . . 88 // 9 // 5.2 Linear equations of the theory of geodesic mappings... 90 // 5.2.1 Mikeš-Berezovski equations of geodesic mappings... 91 // 5.2.2 Linear equations of geodesic mappings of manifolds // with affine connection onto Riemannian manifolds ... 93 // 5.2.3 Example... 96 // 5.3 Geodesic mappings of special manifolds... 97 // 5.3.1 Geodesic mappings of semisymmetric manifolds... 97 // 5.3.2 Geodesic mappings of generalized recurrent manifolds . . 103 // 6 MAPPINGS BETWEEN RIEMANNIAN MANIFOLDS 107
// 6.1 General results on geodesic mappings between Ą„...107 // 6.1.1 Some further properties of geodesic mappings of V„ . . . 107 // 6.1.2 Geodesic mappings of Vn(I?) spaces...108 // 6.2 Classical examples of geodesic mappings...Ill // 6.2.1 Lagrange and Beltrami projections...Ill // 6.2.2 Dimension two...Ill // 6.2.3 Levi-Civita metrics...115 // 6.3 Special mappings and equidistant spaces...115 // 6.3.1 Concircular vector fields and equidistant spaces...115 // 6.3.2 Special mappings for equidistant spaces...116 // 6.3.3 Conformal and affine mappings...117 // 6.3.4 Geodesic mappings between equidistant spaces...118 // 6.3.5 Harmonic mappings between equidistant spaces...119 // 6.3.6 Conformally-projective harmonic mappings...119 // 6.3.7 Equivolume mappings...120 // 7 GEODESIC MAPPINGS // OF SPECIAL RIEMANNIAN MANIFOLDS 121 // 7.1 GM of spaces of constant curvature and Einstein spaces...121 // 7.1.1 Spaces of constant curvature ...121 // 7.1.2 Geodesic mappings of spaces of constant curvature ... 123 // 7.1.3 Geodesic mappings of Einstein spaces ...125 // 7.1.4 Einstein equidistant spaces ...127 // 7.1.5 Geodesic mappings of four-dimensional Einstein spaces . . 127 // 7.1.6 Petrov’s conjecture on geodesic mappings of Einstein spaceslSl // 7.2 Generalized symmetric, recurrent and semisymmetric manifolds 131 // 7.2.1 Generalized symmetric, recurrent and semisymmetric Ą„ . 131 // 7.2.2 GM of semisymmetric spaces and their generalizations . . 132 // 7.2.3 Geodesic mapping
of T-pseudosymmetric spaces ...135 // 7.2.4 Geodesic Mappings of Spaces with Harmonic Curvature . 137 // 7.3 Geodesic mappings of Kahler manifolds...139 // 7.3.1 Introduction...139 // 7.3.2 Equidistant Kahler spaces...139 // 7.3.3 On Sasaki spaces and equidistant Kahler manifolds ... 141 // 7.3.4 Geodesic mappings onto Kahler manifolds...142 // 7.3.5 Geodesic mappings between Kahler manifolds...142 // 10 // 8 GEODESIC DEFORMATIONS // OF HYPERSURFACES IN RIEMANNIAN SPACES 145 // 8.1 Introduction...145 // 8.2 Infinitesimal deformations of Riemannian spaces...145 // 8.3 Geodesic deformations and geodesic maps ...147 // 8.4 Geodesic deformations of subspaces of Riemannian spaces . . . 148 // 8.5 Basic equations of geodesic deformations of hypersurfaces ... 149 // 8.6 A system of equations of Cauchy type // for geodesic deformations of a hypersurface...150 // 9 EXISTENCE OF GLOBAL GEODESIC MAPPINGS // AND PROJECTIVE DEFORMATIONS 155 // 9.1 Projective transformations...155 // 9.2 Global projective transformation of n-sphere...155 // 9.3 Surface of revolution...158 // 9.4 Compact orientable spaces Ln...161 // 9.5 Global geodesic mappings of semisymmetric mappings ...161 // 9.6 Global geodesic mappings Vn onto Vn with boundary...163 // 9.7 On geodesic mappings with certain initial conditions .164 // 9.7.1 The first quadratic integral of a geodesic...164 // 9.7.2 On first quadratic integral of geodesics with special initial // conditions...165 // 10 DIFFERENTIABLE STRUCTURE // ON ELEMENTARY
GEOMETRIES 167 // 10.1 Introduction...167 // 10.2 A Riccati differential equation...168 // 10.3 Differentiable ffi2-planes...169 // 10.4 Generalized shift K2-planes...170 // 10.5 Generalized affine Moulton planes...173 // 10.6 Groups of affine mappings in the Moulton planes A(s,p) . . . 176 // 10.7 Appendix...183 // BIBLIOGRAPHY 185 // SUBJECT INDEX 211 // NAME INDEX // 217
(OCoLC)320227149
cnb001855904

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