Riemannian Geometry in an Orthogonal Frame

by Cartan, Elie; Chern, S. S.; Goldberg, Vladislav V.; Finikov, S. P.

• ISBN13: 9789810247461
• ISBN10: 981024746X
• Format: Hardcover
• Copyright: 2002-05-01
• Publisher: World Scientific Pub Co Inc

Summary

Elie Cartan's book Geometry of Riemannian Manifolds (1928) was one of the best introductions to his methods. It was based on lectures given by the author at the Sorbonne in the academic year 1925-26. A modernized and extensively augmented edition appeared in 1946 (2nd printing, 1951, and 3rd printing, 1988). Cartan's lectures in 1926-27 were different -- he introduced exterior forms at the very beginning and used extensively orthonormal 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. The lectures were translated into Russian in the book Riemannian Geometry in an Orthogonal Frame (1960). 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. The only book of Elie Cartan that wa

Table of Contents

Foreword	v
Translator's Introduction	vii
Preface to the Russian Edition	ix
PRELIMINARIES	1	(34)
Method of Moving Frames	3	(6)
Components of an infinitesimal displacement	3	(1)
Relations among 1-forms of an orthonormal frame	4	(1)
Finding the components of a given family of trihedrons	5	(1)
Moving frames	5	(1)
Line element of the space	6	(1)
Contravariant and covariant components	7	(1)
Infinitesimal affine transformations of a frame	8	(1)
The Theory of Pfaffian Forms	9	(10)
Differentiation in a given direction	9	(1)
Bilinear covariant of Frobenius	10	(3)
Skew-symmetric bilinear forms	13	(1)
Exterior quadratic forms	14	(1)
Converse theorems. Cartan's Lemma	15	(3)
Exterior differential	18	(1)
Integration of Systems of Pfaffian Differential Equations	19	(10)
Integral manifold of a system	19	(1)
Necessary condition of complete integrability	20	(1)
Necessary and sufficient condition of complete integrability of a system of Pfaffian equations	21	(1)
Path independence of the solution	22	(2)
Reduction of the problem of integration of a completely integrable system to the integration of a Cauchy system	24	(1)
First integrals of a completely integrable system	25	(1)
Relation between exterior differentials and the Stokes formula	25	(2)
Orientation	27	(2)
Generalization	29	(6)
Exterior differential forms of arbitrary order	29	(2)
The Poincare theorem	31	(1)
The Gauss formula	31	(2)
Generalization of Theorem 6 of No. 12	33	(2)
A. GEOMETRY OF EUCLIDEAN SPACE	35	(46)
The Existence Theorem for a Family of Frames with Given Infinitesimal Components ωi and ωij	37	(8)
Family of oblique trihedrons	37	(1)
The family of orthonormal tetrahedrons	38	(1)
Family of oblique trihedrons with a given line element	39	(1)
Integration of system (I) by the method of the form invariance	40	(1)
Particular cases	41	(2)
Spaces of trihedrons	43	(2)
The Fundamental Theorem of Metric Geometry	45	(10)
The rigidity of the point space	45	(1)
Geometric meaning of the Weyl theorem	46	(1)
Deformation of the tangential space	47	(3)
Deformation of the plane considered as a locus of straight lines	50	(2)
Ruled space	52	(3)
Vector Analysis in an n-Dimensional Euclidean Space	55	(12)
Transformation of the space with preservation of a line element	55	(3)
Equivalence of reduction of a line element to a sum of squares to the choosing of a frame to be orthogonal	58	(1)
Congruence and symmetry	59	(1)
Determination of forms ωji for given forms ωi	60	(1)
Three-dimensional case	61	(1)
Absolute differentiation	62	(2)
Divergence of a vector	64	(1)
Differential parameters	65	(2)
The Fundamental Principles of Tensor Algebra	67	(8)
Notion of a tensor	67	(2)
Tensor algebra	69	(2)
Geometric meaning of a skew-symmetric tensor	71	(2)
Scalar product of a bivector and a vector and of two bivectors	73	(1)
Simple rotation of a rigid body around a point	74	(1)
Tensor Analysis	75	(6)
Absolute differentiation	75	(1)
Rules of absolute differentiation	76	(1)
Exterior differential tensor-valued form	77	(1)
A problem of absolute exterior differentiation	78	(3)
B. THE THEORY OF RIEMANNIAN MANIFOLDS	81	(36)
The Notion of a Manifold	83	(4)
The general notion of a manifold	83	(1)
Analytic representation	84	(1)
Riemannian manifolds Regular metric	84	(3)
Locally Euclidean Riemannian Manifolds	87	(6)
Definition of a locally Euclidean manifold	87	(1)
Examples	87	(2)
Riemannian manifold with an everywhere regular metric	89	(1)
Locally compact manifold	89	(1)
The holonomy group	90	(1)
Discontinuity of the holonomy group of the locally Euclidean manifold	91	(2)
Euclidean Space Tangent at a Point	93	(8)
Euclidean tangent metric	93	(1)
Tangent Euclidean space	94	(2)
The main notions of vector analysis	96	(2)
Three methods of introducing a connection	98	(1)
Euclidean metric osculating at a point	99	(2)
Osculating Euclidean Space	101	(10)
Absolute differentiation of vectors on a Riemannian manifold	101	(1)
Geodesics of a Riemannian manifold	102	(1)
Generalization of the Frenet formulas. Curvature and torsion	103	(1)
The theory of curvature of surfaces in a Riemannian manifold	104	(2)
Geodesic torsion. The Enneper theorem	106	(2)
Conjugate directions	108	(1)
The Dupin theorem on a triply orthogonal system	108	(3)
Euclidean Space of Conjugacy along a Line	111	(6)
Development of a Riemannian manifold in Euclidean space along a curve	111	(1)
The constructed representation and the osculating Euclidean space	112	(1)
Geodesics. Parallel surfaces	113	(2)
Geodesics on a surface	115	(2)
C. CURVATURE AND TORSION OF A MANIFOLD	117	(50)
Space with a Euclidean Connection	119	(14)
Determination of forms ωji for given forms ωi	119	(2)
Condition of invariance of line element	121	(2)
Axioms of equipollence of vectors	123	(1)
Space with Euclidean connection	124	(1)
Euclidean space of conjugacy	125	(1)
Absolute exterior differential	126	(1)
Torsion of the manifold	127	(1)
Structure equations of a space with Euclidean connection	128	(1)
Translation and rotation associated with a cycle	129	(1)
The Bianchi identities	130	(1)
Theorem of preservation of curvature and torsion	130	(3)
Riemannian Curvature of a Manifold	133	(16)
The Bianchi identities in a Riemannian manifold	133	(1)
The Riemann-Christoffel tensor	134	(2)
Riemannian curvature	136	(1)
The case n = 2	137	(2)
The case n = 3	139	(1)
Geometric theory of curvature of a three-dimensional Riemannian manifold	140	(1)
Schur's theorem	141	(1)
Example of a Riemannian space of constant curvature	142	(2)
Determination of the Riemann-Christoffel tensor for a Riemannian curvature given for all planar directions	144	(1)
Isotropic n-dimensional manifold	145	(1)
Curvature in two different two-dimensional planar directions	146	(1)
Riemannian curvature in a direction of arbitrary dimension	146	(1)
Ricci tensor. Einstein's quadric	147	(2)
Spaces of Constant Curvature	149	(8)
Congruence of spaces of the same constant curvature	149	(2)
Existence of spaces of constant curvature	151	(1)
Proof of Schur	152	(1)
The system is satisfied by the solution constructed	153	(4)
Geometric Construction of a Space of Constant Curvature	157	(10)
Spaces of constant positive curvature	157	(2)
Mapping onto an n-dimensional projective space	159	(1)
Hyperbolic space	160	(1)
Representation of vectors in hyperbolic geometry	161	(1)
Geodesics in Riemannian manifold	161	(1)
Pseudoequipollent vectors: pseudoparallelism	162	(2)
Geodesics in spaces of constant curvature	164	(2)
The Cayley metric	166	(1)
D. THE THEORY OF GEODESIC LINES	167	(26)
Variational Problems for Geodesics	169	(10)
The field of geodesics	169	(1)
Stationary state of the arc length of a geodesic in the family of lines joining two points	170	(1)
The first variation of the arc length of a geodesic	171	(1)
The second variation of the arc length of a geodesic	172	(1)
The minimum for the arc length of a geodesic (Darboux's proof)	173	(1)
Family of geodesics of equal length intersecting the same geodesic at a constant angle	174	(5)
Distribution of Geodesics near a Given Geodesic	179	(10)
Distance between neighboring geodesics and curvature of a manifold	179	(2)
The sum of the angles of a parallelogramoid	181	(1)
Stability of a motion of a material system without external forces	182	(1)
Investigation of the maximum and minimum for the length of a geodesic in the case Aij = const.	183	(2)
Symmetric vectors	185	(1)
Parallel transport by symmetry	186	(1)
Determination of three-dimensional manifolds, in which the parallel transport preserves the curvature	187	(2)
Geodesic Surfaces	189	(4)
Geodesic surface at a point. Severi's method of parallel transport of a vector	189	(1)
Totally geodesic surfaces	190	(1)
Development of lines of a totally geodesic surface on a plane	191	(1)
The Ricci theorem on orthogonal trajectories of totally geodesic surfaces	191	(2)
E. EMBEDDED MANIFOLDS	193	(60)
Lines in a Riemannian Manifold	195	(16)
The Frenet formulas in a Riemannian manifold	195	(1)
Determination of a curve with given curvature and torsion. Zero torsion curves in a space of constant curvature	196	(3)
Curves with zero torsion and constant curvature in a space of constant negative curvature	199	(4)
Integration of Frenet's equations of these curves	203	(2)
Euclidean space of conjugacy	205	(1)
The curvature of a Riemannian manifold occurs only in infinitesimals of second order	206	(5)
Surfaces in a Three-Dimensional Riemannian Manifold	211	(10)
The first two structure equations and their geometric meaning	211	(1)
The third structure equation. Invariant forms (scalar and exterior)	212	(1)
The second fundamental form of a surface	213	(2)
Asymptotic lines. Euler's theorem. Total and mean curvature of a surface	215	(1)
Conjugate tangents	216	(1)
Geometric meaning of the form ψ	216	(1)
Geodesic lines on a surface. Geodesic torsion. Enneper's theorem	217	(4)
Forms of Laguerre and Darboux	221	(18)
Laguerre's form	221	(1)
Darboux's form	222	(2)
Riemannian curvature of the ambient manifold	224	(1)
The second group of structure equations	225	(1)
Generalization of classical theorems on normal curvature and geodesic torsion	226	(2)
Surfaces with a given line element in Euclidean space	228	(1)
Problems on Laguerre's form	229	(2)
Invariance of normal curvature under parallel transport of a vector	231	(2)
Surfaces in a space of constant curvature	233	(6)
p-Dimensional Submanifolds in a Riemannian Manifold of n Dimensions	239	(14)
Absolute variation of a tangent vector. Inner differentiation. Euler's curvature	239	(1)
Tensor character of Euler's curvature	240	(1)
The second system of structure equations	241	(1)
Particular Cases
A Hypersurface in a Four-dimensional Space
Principal directions and principal curvatures	242	(1)
Hypersurface in the Euclidean space	243	(1)
Ellipse of normal curvature	244	(3)
Two-dimensional Surfaces in a Four-dimensional Manifold
Generalization of classical notions	247	(1)
Minimal surfaces	248	(1)
Finding minimal surfaces	249	(4)
Index	253