Riemannian Geometry in an Orthogonal Frame : From Lectures Delivered by Elie Cartan at the Sorbonne in 1926-1927.
Title:
Riemannian Geometry in an Orthogonal Frame : From Lectures Delivered by Elie Cartan at the Sorbonne in 1926-1927.
Author:
Cartan, Ã.
ISBN:
9789812799715
Personal Author:
Physical Description:
1 online resource (278 pages)
Contents:
Contents -- Foreword -- Translator's Introduction -- Preface to the Russian Edition -- PRELIMINARIES -- Chapter 1 Method of Moving Frames -- 1. Components of an infinitesimal displacement -- 2. Relations among 1-forms of an orthonormal frame -- 3. Finding the components of a given family of trihedrons -- 4. Moving frames -- 5. Line element of the space -- 6. Contravariant and covariant components -- 7. Infinitesimal affine transformations of a frame -- Chapter 2 The Theory of Pfaffian Forms -- 8. Differentiation in a given direction -- 9. Bilinear covariant of Frobenius -- 10. Skew-symmetric bilinear forms -- 11. Exterior quadratic forms -- 12. Converse theorems. Cartan's Lemma -- 13. Exterior differential -- Chapter 3 Integration of Systems of Pfaffian Differential Equations -- 14. Integral manifold of a system -- 15. Necessary condition of complete integrability -- 16. Necessary and sufficient condition of complete integrability of a system of Pfaffian equations -- 17. Path independence of the solution -- 18. Reduction of the problem of integration of a completely integrable system to the integration of a Cauchy system -- 19. First integrals of a completely integrable system -- 20. Relation between exterior differentials and the Stokes formula -- 21. Orientation -- Chapter 4 Generalization -- 22. Exterior differential forms of arbitrary order -- 23. The Poincare theorem -- 24. The Gauss formula -- 25. Generalization of Theorem 6 of No. 12 -- A. GEOMETRY OF EUCLIDEAN SPACE -- Chapter 5 The Existence Theorem for a Family of Frames with Given Infinitesimal Components wi and wji -- 26. Family of oblique trihedrons -- 27. The family of orthonormal tetrahedrons -- 28. Family of oblique trihedrons with a given line element.
29. Integration of system (I) by the method of the form invariance -- 30. Particular cases -- 31. Spaces of trihedrons -- Chapter 6 The Fundamental Theorem of Metric Geometry -- 32. The rigidity of the point space -- 33. Geometric meaning of the Weyl theorem -- 34. Deformation of the tangential space -- 35. Deformation of the plane considered as a locus of straight lines -- 36. Ruled space -- Chapter 7 Vector Analysis in an n-Dimensional Euclidean Space -- 37. Transformation of the space with preservation of a line element -- 38. Equivalence of reduction of a line element to a sum of squares to the choosing of a frame to be orthogonal -- 39. Congruence and symmetry -- 40. Determination of forms wji for given forms wi -- 41. Three-dimensional case -- 42. Absolute differentiation -- 43. Divergence of a vector -- 44. Differential parameters -- Chapter 8 The Fundamental Principles of Tensor Algebra -- 45. Notion of a tensor -- 46. Tensor algebra -- 47. Geometric meaning of a skew-symmetric tensor -- 48. Scalar product of a bivector and a vector and of two bivectors -- 49. Simple rotation of a rigid body around a point -- Chapter 9 Tensor Analysis -- 50. Absolute differentiation -- 51. Rules of absolute differentiation -- 52. Exterior differential tensor-valued form -- 53. A problem of absolute exterior differentiation -- B. THE THEORY OF RIEMANNIAN MANIFOLDS -- Chapter 10 The Notion of a Manifold -- 54. The general notion of a manifold -- 55. Analytic representation -- 56. Riemannian manifolds. Regular metric -- Chapter 11 Locally Euclidean Riemannian Manifolds -- 57. Definition of a locally Euclidean manifold -- 58. Examples -- 59. Riemannian manifold with an everywhere regular metric -- 60. Locally compact manifold -- 61. The holonomy group.
62. Discontinuity of the holonomy group of the locally Euclidean manifold -- Chapter 12 Euclidean Space Tangent at a Point -- 63. Euclidean tangent metric -- 64. Tangent Euclidean space -- 65. The main notions of vector analysis -- 66. Three methods of introducing a connection -- 67. Euclidean metric osculating at a point -- Chapter 13 Osculating Euclidean Space -- 68. Absolute differentiation of vectors on a Riemannian manifold -- 69. Geodesics of a Riemannian manifold -- 70. Generalization of the Frenet formulas. Curvature and torsion -- 71. The theory of curvature of surfaces in a Riemannian manifold -- 72. Geodesic torsion. The Enneper theorem -- 73. Conjugate directions -- 74. The Dupin theorem on a triply orthogonal system -- Chapter 14 Euclidean Space of Conjugacy along a Line -- 75. Development of a Riemannian manifold in Euclidean space along a curve -- 76. The constructed representation and the osculating Euclidean space -- 77. Geodesics. Parallel surfaces -- 78. Geodesics on a surface -- C. CURVATURE AND TORSION OF A MANIFOLD -- Chapter 15 Space with a Euclidean Connection -- 79. Determination of forms wji for given forms wi -- 80. Condition of invariance of line element -- 81. Axioms of equipollence of vectors -- 82. Space with Euclidean connection -- 83. Euclidean space of conjugacy -- 84. Absolute exterior differential -- 85. Torsion of the manifold -- 86. Structure equations of a space with Euclidean connection -- 87. Translation and rotation associated with a cycle -- 88. The Bianchi identities -- 89. Theorem of preservation of curvature and torsion -- Chapter 16 Riemannian Curvature of a Manifold -- 90. The Bianchi identities in a Riemannian manifold -- 91. The Riemann-Christoffel tensor -- 92. Riemannian curvature -- 93. The case n = 2 -- 94. The case n = 3.
95. Geometric theory of curvature of a three-dimensional Riemannian manifold -- 96. Schur's theorem -- 97. Example of a Riemannian space of constant curvature -- 98. Determination of the Riemann-Christoffel tensor for a Riemannian curvature given for all planar directions -- 99. Isotropic n-dimensional manifold -- 100. Curvature in two different two-dimensional planar directions -- 101. Riemannian curvature in a direction of arbitrary dimension -- 102. Ricci tensor. Einstein's quadric -- Chapter 17 Spaces of Constant Curvature -- 103. Congruence of spaces of the same constant curvature -- 104. Existence of spaces of constant curvature -- 105. Proof of Schur -- 106. The system is satisfied by the solution constructed -- Chapter 18 Geometric Construction of a Space of Constant Curvature -- 107. Spaces of constant positive curvature -- 108. Mapping onto an n-dimensional projective space -- 109. Hyperbolic space -- 110. Representation of vectors in hyperbolic geometry -- 111. Geodesics in Riemannian manifold -- 112. Pseudoequipollent vectors: pseudoparallelism -- 113. Geodesics in spaces of constant curvature -- 114. The Cayley metric -- D. THE THEORY OF GEODESIC LINES -- Chapter 19 Variational Problems for Geodesics -- 115. The field of geodesics -- 116. Stationary state of the arc length of a geodesic in the family of lines joining two points -- 117. The first variation of the arc length of a geodesic -- 118. The second variation of the arc length of a geodesic -- 119. The minimum for the arc length of a geodesic (Darboux's proof) -- 120. Family of geodesics of equal length intersecting the same geodesic at a constant angle -- Chapter 20 Distribution of Geodesics near a Given Geodesic -- 121. Distance between neighboring geodesics and curvature of a manifold.
122. The sum of the angles of a parallelogramoid -- 123. Stability of a motion of a material system without external forces -- 124. Investigation of the maximum and minimum for the length of a geodesic in the case Aij = const. -- 125. Symmetric vectors -- 126. Parallel transport by symmetry -- 127. Determination of three-dimensional manifolds in which the parallel transport preserves the curvature -- Chapter 21 Geodesic Surfaces -- 128. Geodesic surface at a point. Severi's method of parallel transport of a vector -- 129. Totally geodesic surfaces -- 130. Development of lines of a totally geodesic surface on a plane -- 131. The Ricci theorem on orthogonal trajectories of totally geodesic surfaces -- E. EMBEDDED MANIFOLDS -- Chapter 22 Lines in a Riemannian Manifold -- 132. The Frenet formulas in a Riemannian manifold -- 133. Determination of a curve with given curvature and torsion. Zero torsion curves in a space of constant curvature -- 134. Curves with zero torsion and constant curvature in a space of constant negative curvature -- 135. Integration of Frenet's equations of these curves -- 136. Euclidean space of conjugacy -- 137. The curvature of a Riemannian manifold occurs only in infinitesimals of second order -- Chapter 23 Surfaces in a Three-Dimensional Riemannian Manifold -- 138. The first two structure equations and their geometric meaning -- 139. The third structure equation. Invariant forms (scalar and exterior) -- 140. The second fundamental form of a surface -- 141. Asymptotic lines. Euler's theorem. Total and mean curvature of a surface -- 142. Conjugate tangents -- 143. Geometric meaning of the form w -- 144. Geodesic lines on a surface. Geodesic torsion. Enneper's theorem -- Chapter 24 Forms of Laguerre and Darboux -- 145. Laguerre's form -- 146. Darboux's form.
147. Riemannian curvature of the ambient manifold.
Abstract:
Foreword by S S Chern. In 1926-27, Cartan gave a series of lectures in which he introduced exterior forms at the very beginning and used extensively orthogonal frames throughout to investigate the geometry of Riemannian manifolds. In this course he solved a series of problems in Euclidean and non-Euclidean spaces, as well as a series of variational problems on geodesics. In 1960, Sergei P Finikov translated from French into Russian his notes of these Cartan's lectures and published them as a book entitled Riemannian Geometry in an Orthogonal Frame. This book has many innovations, such as the notion of intrinsic normal differentiation and the Gaussian torsion of a submanifold in a Euclidean multidimensional space or in a space of constant curvature, an affine connection defined in a normal fiber bundle of a submanifold, etc. It has now been translated into English by Vladislav V Goldberg, currently Distinguished Professor of Mathematics at the New Jersey Institute of Technology, USA, who also edited the Russian edition. Contents: Method of Moving Frames; The Theory of Pfaffian Forms; Integration of Systems of Pfaffian Differential Equations; Generalization; The Existence Theorem for a Family of Frames with Given Infinitesimal Components ω i and ω i j; The Fundamental Theorem of Metric Geometry; Vector Analysis in an n -Dimensional Euclidean Space; The Fundamental Principles of Tensor Algebra; Tensor Analysis; The Notion of a Manifold; Locally Euclidean Riemannian Manifolds; Euclidean Space Tangent at a Point; Osculating Euclidean Space; Euclidean Space of Conjugacy Along a Line; Space with a Euclidean Connection; Riemannian Curvature of a Manifold; Spaces of Constant Curvature; Geometric Construction of a Space of Constant Curvature; Variational Problems for Geodesics; Distribution of Geodesics Near a Given Geodesic; Geodesic Surfaces; Lines in a
Riemannian Manifold; Surfaces in a Three-Dimensional Riemannian Manifold; Forms of Laguerre and Darboux; p -Dimensional Submanifolds in a Riemannian Manifold of n Dimensions. Readership: Senior undergraduates, graduate students and researchers in geometry and topology.
Local Note:
Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2017. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.
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