This volume of the Encyclopaedia contains two articles which give a survey of modem research into non-regular Riemannian geometry, carried out mostly by Russian mathematicians. The first article written by Reshetnyak is devoted to the theory of two-dimensional Riemannian manifolds of bounded curvature. Concepts of Riemannian geometry such as the area and integral curvature of a set and the length and integral curvature of a curve are also defined for these manifolds. Some fundamental results of Riemannian geometry like the Gauss-Bonnet formula are true in the more general case considered in the book. The second article by Berestovskij and Nikolaev is devoted to the theory of metric spaces whose curvature lies between two given constants. The main result is that these spaces are in fact Riemannian. This result has important applications in global Riemannian geometry. Both parts cover topics which have not yet been treated in monograph form. Hence the book will be immensely useful to graduate students and researchers in geometry, in particular Riemannian geometry.
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目录
Ⅰ Two-Dimensional Manifolds of Bounded Curvature Chapter 1 Preliminary Information §1. Introduction 1.1. General Information about the Subject of Research and a Survey of Results 1.2. Some Notation and Terminology §2. The Concept of a Space with Intrinsic Metric 2.1. The Concept of the Length of a Parametrized Curve 2.2. A Space with Intrinsic Metric.The Induced Metric 2.3. The Concept of a Shortest Curve 2.4. The Operation of Cutting of a Space with Intrinsic Metric §3. Two-Dimensional Manifolds with Intrinsic Metric 3.1. Definition.Triangulation of a Manifold 3.2. Pasting of Two-Dimensional Manifolds with Intrinsic Metric 3.3. Cutting of Manifolds 3.4. A Side of a Simple Arc in a Two-Dimensional Manifold §4. Two-Dimensional Riemannian Geometry 4.1. Differentiable Two-Dimensional Manifolds 4.2. The Concept of a Two-Dimensional Riemannian Manifold 4.3. The Curvature of a Curve in a Riemannian Manifold.Integral Curvature.The Gauss-Bonnet Formula 4.4. Isothermal Coordinates in Two-Dimensional Riemannian Manifolds of Bounded Curvature §5. Manifolds with Polyhedral Metric 5.1. Cone and Angular Domain 5.2. Definition of a Manifold with Polyhedral Metric 5.3. Curvature of a Set on a Polyhedron.Turn of the Boundary.I'he Gauss-Bonnet Theorem 5.4. A Turn of a Polygonal Line on a Polyhedron 5.5. Characterization of the Intrinsic Geometry of Convex Polyhedra 5.6. An Extremal Property of a Convex Cone.The Method of Cutting and Pasting as a Means of Solving Extremal Problems for Polyhedra 5.7. The Concept of a K-Polyhedron Chapter 2 Different Ways of Defining Two-Dimensional Manifolds of Bounded Curvature §6. Axioms of a Two-Dimensional Manifold of Bounded Curvature.Characterization of such Manifolds by Means of Approximation by Polyhedra 6.1. Axioms of a Two-Dimensional Manifold of Bounded Curvature 6.2. Theorems on the Approximation of Two-Dimensional Manifolds of Bounded Curvature by Manifolds with Polyhedral and Riemannian Metric 6.3. Proof of the First Theorem on Approximation 6.4. Proof of Lemma 6.3.1 6.5. Proof of the Second Theorem on Approximation §7. Analytic Characterization of Two-Dimensional Manifolds of Bounded Curvature 7.1. Theorems on Isothermal Coordinates in a Two-Dimensional Manifold of Bounded Curvature 7.2. Some Information about Curves on a Plane and in a Riemannian manifold 7.3. Proofs of Theorems 7.1.1,7.1.2,7.1.3 7.4. On the ProofofTheorem 7.3.1 Chapter 3 Basic Facts of the Theory of Manifolds of Bounded Curvature §8. Basic Results of the Theory of Two-Dimensional Manifolds of Bounded Curvature 8.1. A Turn of a Curve and the Integral Curvature of a Set 8.2. A Theorem on the Contraction of a Cone.Angle between Curves.Comparison Theorems 8.3. A Theorem on Pasting Together Two-Dimensional Manifolds of Bounded Curvature 8.4. Theorems on Passage to the Limit for Two-Dimensional Manifolds of Bounded Curvature 8.5. Some Inequalities and Estimates.Extremal Problems for Two-Dimensional Manifolds of Bounded Curvature §9. Further Results.Some Additional Remarks References Ⅱ Multidimensional Generalized Riemannian Spaces Introduction 0.1. RiemannianSpaces 0.2. Generalized Riemannian Spaces 0.3. Riemannian Geometry and Generalized Riemannian Spaces 0.4. A Brief Characterization of the Article by Chapters 0.5. In What Sense Do the Stated Results Have Multidimensional Character? 0.6. Final Remarks on the Text Chapter 1 Basic Concepts Connected with the Intrinsic Metric §1 Intrinsic Metric.Shortest Curve.Triangle.Angle.Excess of a Triangle 1.1. Intrinsic Metric 1.2. Shortest Curve 1.3. Triangle 1.4. Angle 1.5. Excess §2. General Propositions about Upper Angles §3. The Space of Directions at a Point.K-Cone.Tangent Space 3.1. Direction 3.2. K-Cone 3.3. Tangent Space §4. Remarks.Examples 4.1. Intrinsic Metric.Shortest Curve.Angles 4.2. An Assertion Completely Dual to Theorem 2.1,that is.the Corresponding Lower Bound for the Lower Angle Does not Hold 4.3. Tangent Space.Space of Directions Chapter 2 Spaces of Curvature ≤ K (and ≥K') §5. Spaces of Curvature ≤ K.The Domain RK,and its Basic Properties 5.1. Definition of a Space of Curvature ≤ K 5.2. Basic Properties of the Domain RK 5.3. The Domain PK §6. The Operation of Gluing 6.1. Gluing of Metric Spaces with Intrinsic Metric §7. Equivalent Definitions of Upper Boundedness of the Curvature 7.1. Conditions under which a Space of Curvature ≤ K is the Domain RK 7.2. Connection with the Riemannian Definition of Curvature 7.3. Definition of Upper Boundedness of Curvature 7.4. Non-Expanding Maps in Spaces of Curvature ≤ K 7.5. Boundedness of the Curvature from the Viewpoint of Distance Geometry §8. Space of Directions,Tangent Space at a Point of a Space of Curvature ≤ K 8.1. Conditions under which a Shortest Curve Goes out in each Direction 8.2. Intrinsic Metric in Ωp 8.3. Tangent Space §9. Surfaces and their Areas 9.1. The Definition of the Area of a Surface 9.2. Properties of Area 9.3. Ruled Surfaces in RK 9.4. Isoperimetric Inequality 9.5. Plateau's Problem §10. Spaces of Curvature both ≤ K and ≥ K' 10.1. Definition of a Space of Curvature both ≤ K and ≥ K' 10.2. Basic Properties of a Domain RK'.K 10.3. Equivalent Definitions of Boundedness of Curvature §11. Remarks,Examples 11.1. Spaces of Curvature ≤ K as a Generalization of Riemannian Spaces 11.2. Polyhedral Metrics 11.3. Spaces of Curvature ≥ K' Chapter 3 Spaces with Bounded Curvature §12. CO-RiemannianS tructure in Spaces with Bounded Curvature 12.1. Definition of a Space with Bounded Curvature 12.2. The Tangent Space at a Point of a Space with Bounded Curvature 12.3. Introduction of CO-Smooth Riemannian Structure §13. Parallel Translation in Spaces with Bounded Curvature 13.1. Construction of a Parallel Translation 13.2. Statement of the Main Results 13.3. Plan of the Proof of the Main Results of §13 §14. Smoothness of the Metric of Spaces with Bounded Curvature 14.1. Statement of the Main Result 14.2. Plan of the Proof of Theorem 14.1 §15. Spaces with Bounded Curvature and Limits of Smooth Riemannian Metrics 15.1. Approximation of the Metric of a Space with Bounded Curvature by Smooth Riemannian Metrics 15.2. A Space of Riemannian Manifolds with Sectional Curvatures Bounded in Aggregate Chapter 4 Existence of the Curvature of a Metric Space at a Point and the Axioms of Riemannian Geometry §16. The Space of Directions of an Arbitrary Metric Space 16.1. Distance between Directions 16.2. Space of Directions §17. Curvature of a Metric Space 17.1. Definition of Non-isotropic Riemannian Curvature 17.2. Existence of Curvature at a Point 17.3. Geometrical Meaning of Sectional Curvature 17.4. Isotropic Riemannian Curvature 17.5. Wald Curvature and its Connection with Isotropic Riemannian Curvature 17.6. Continuity of Curvature §18. Axioms of Riemannian Geometry 18.1. Synthetic Description of C2,α-Smooth Riemannian Manifolds 18.2. Synthetic Description of Cm,α-Smooth Riemannian Manifolds(m=3,4,...) 18.3. Isotropic Metric Spaces References Author Index Subject Index
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