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Geometric 4 Informal Riemannian Geometry

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geometric 4 informal riemannian geometry
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Geometric 4 (Informal Riemannian Geometry), by Yu.G.Reshetnyak, English, Hardcover, 2009

Product Details



Press Science Press

Isbn 7030235010/9787030235015

Publication Date Jan 2009

Pages 250

Size 16K

Edition First

Cover Hardcover

Language English

Page material gelatine plate paper

Product Description:

This volume of the Encyclopaedia contains two articles which give a survey of modern 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 giyen 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.


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. The 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 Proof of Theorem 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



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