The theory of surfaces in Euclidean spaces is remarkably rich in deep results and applications. This volume of the Encyclopaedia is concerned mainly with the connection between the theory of embedded surfaces and Riemannian geometry and with the geometry of surfaces as influenced bv intrinsic metrics.
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目录
I. The Geometry of Surfaces in Euclidean Spaces Preface Chapter 1 The Geometry of Two-Dimensional Manifolds and Surfaces in En §1. Statement of the Problem 1.1. Classes of Metrics and Classes of Surfaces. Geometric Groups and Geometric Properties §2. Smooth Surfaces 2.1. Types of Points 2.2. Classes of Surfaces 2.3. Classes of Metrics 2.4 G-Connectedness 2.5. Results and Conjectures 2.6. The Conformal Group §3. Convex, Saddle and Developable Surfaces with No Smoothness Requirement 3.1. Classes of Non-Smooth Surfaces and Metrics 3.2. Questions of Approximation 3.3. Results and Conjectures §4. Surfaces and Metrics of Bounded Curvature 4.1. Manifolds of Bounded Curvature 4.2. Surfaces of Bounded Extrinsic Curvature Chapter 2 Convex Surfaces §1. Weyl's Problem 1.1. Statement of the Problem 1.2. Historical Remarks 1.3. Outline of One of the Proofs §2. The Intrinsic Geometry of Convex Surfaces. The Generalized Weyl Problem 2.1. Manifolds of Non-Negative Curvat.ure in the Sense of Aleksandrov 2.2. Solution of the Generalized Weyl Problem 2.3. The Gluing Theorem §3. Smoothness of Convex Surfaces 3.1. Smoothness of Convex Immersions 3.2. The Advantage of Isothermal Coordinates 3.3. Consequences of the Smoothness Theorems §4. Bendings of Convex Surfaces 4.1. Basic Concepts 4.2. Smoothness of Bendings 4.3. The Existence of Bendings 4.4. Connection Between Different Forms of Bendings §5. Unbendability of Closed Convex Surfaces 5.1. Unique Determination 5.2. Stability in Weyl's Problem 5.3. Use of the Bending Field §6. Infinite Convex Surfaces 6.1 Non-Compact Surfaces 6.2. Description of Bendings §7. Convex Surfaces with Given Curvatures 7.1. Hypersurfaces 7.2. Minkowski's Problem 7.3. Stability 7.4. Curvature Functions and Analogues of the Minkowski Problem 7.5. Connection with the Monge-Ampère Equations §8. Individual Questions of the Connection Between the Intrinsic and Extrinsic Geometry of Convex Surfaces 8.1. Properties of Surfaces 8.2. Properties of Curves 8.3. The Spherical Image of a Shortest Curve 8.4. The Possibility of Certain Singularities Vanishing Under Bendings Chapter 3 Saddle Surfaces §1. Efimov's Theorem and Conjectures Associated with It 1.1. Sufficient Criteria for Non-Immersibility in E3 1.2. Sufficient Criteria for Immersibility in E3 1.3. Conjecture About a Saddle Immersion in En 1.4. The Possibility of Non-Immersibility when the Manifold is Not Simply-Connected §2. On the Extrinsic Geometry of Saddle Surfaces 2.1. The Variety of Saddle Surfaces 2.2. Tapering Surfaces §3 Non-Regular Saddle Surfaces 3.1. Definitions 3.2. Intrinsic Geometry 3.3. Problems of Immersibility 3.4. Problems of Non-Immersibility Chapter 4 Surfaces of Bounded Extrinsic Curvature §1. Surfaces of Bounded Positive Extrinsic Curvature 1.1. Extrinsic Curvatures of a Smooth Surface 1.2. Extrinsic Curvatures of a General Surface 1.3. Inequalities §2. The Role of the Mean Curvature 2.1. The Mean Curvature of a Non-Smooth Surface 2.2. Surfaces of Bounded Mean Curvature 2.3. Mean Curvature as First Variation of the Area §3. C1-Smooth Surfaces of Bounded Extrinsic Curvature 3.1. The Role of the Condition of Boundedness of the Extrinsic Curvature 3.2. Normal C1-Smooth Surfaces 3.3. The Main Results 3.4. Gauss's Theorem 3.5. C1,α-Smooth Surfaces §4. Polyhedra 4.1. The Role of Polyhedra in the General Theor 4.2. Polyhedral Metric and Polyhedral Surface 4.3. Results and Conjectures §5. Appendix. Smoothness Classes Comments on the References References II. Surfaces of Negative Curvature Preface §1. Hilbert's Theorem 1.1. Statement of the Problem 1.2. Plan of the Proof of Hilbert's Theorem 1.3. Connection with the Equations of Mathematical Physics 1.4. Generalizations §2. Surfaces of Negative Curvature in E3. Examples. Intrinsic and Extrinsic Curvature. Hadamard's Problem 2.1. Examples of Surfaces of Negative Curvature in E3, and Their Extrinsic and Intrinsic Geometry 2.2. Some Remarks on a C1-Isometric Embedding of L2 in E3 According to Kuiper 2.3. Hadamard's Conjecture 2.4. Surfaces of Negative Intrinsic and Bounded Extrinsic Curvature in E3 §3. Surfaces of the Form z=f(x,y); Plan of the Proof of Efimov's Theorem 3.1. Some Results on Surfaces that Project One-to-one on the Plane E2 3.2. A Theorem of Efimov and Heinz on the Extent of a One-to-one Projection onto the Plane of a Surface with Negative Curvature Separated from Zero 3.3. Plan of the Proof of Theorem B §4. Surfaces with Slowly Varying Curvature. Immersion of Metrics of Negative Curvature in E3. The Influence of the Metric on the Regularity of a Surface 4.1. Analytic Apparatus 4.2. (h, △)-Metrics 4.3. q-Metrics 4.4. Immersion of Metrics of Negative Curvature in E3 4.5. Study of the Boundary of a Surface 4.6. Surfaces with Slowly Varying Curvature in a Riemannian Space 4.7. Influence of the Metric on the Regularity of a Surface §5. On Surfaces with a Metric of Negative Curvature in Multidimensional Euclidean Spaces 5.1. Bieberbach's Theorem 5.2. Embedding and Immersion of Lp in EN 5.3. Piecewise-Analytic Immersion of L2 in E4 5.4. Some Results on Non-Immersibility in the Multidimensional Case 5.5. On Closed Surfaces of Negative Curvature Commentary on the References References III. Local Theory of Bendings of Surfaces Preface §1. Definitions and Terminologies 1.1. A Surface and Its Metric 1.2. Isometric Surfaces and Isometric Immersions 1.3. Bendings of Surfaces 1.4. Infinitesimal Bendings of Surfaces 1.5. Bendings of Surfaces and the Theory of Elastic Shells 1.6. Areal Deformations §2. Statement of Problems §3. Connection Between Bendings and Infinitesimal Bendings of Surfaces 3.1. General Equations of Infinitesimal Bendings of Arbitrary Order 3.2. Transition from Infinitesimal Bendings of High Order to infinitesimal Bendings of Low Order 3.3. Transition from Infinitesimal Bendings of Low Order to Infinitesimal Bendings of High Order 3.4. Algebraic Properties of Fields of Infinitesimal Bendings of the lst Order §4. Bendings of Surfaces in the Class C1 §5. Auxiliary Information: Classification and Integral Characteristics of Points of a Surface; Equations of Immersion and Bending 5.1. Four Types of Points on a Surface 5.2. Arithmetic Characteristics of a Regular Point of a Surface 5.3. Stability and Instability of Arithmetic Characteristics of a Point of a Surface 5.4. Equations of an Immersion and a Bending of a Surface §6. Bendings of Surfaces in a Neighbourhood of a Point of General Position 6.1. Analytic Case 6.2. Surfaces of Positive Curvature 6.3. Surfaces of Negative Curvature 6.4. Neighbourhood of a Parabolic Point §7. Bendability of Surfaces with a Flat Point 7.1. Non-Applicable Isometric Surfaces 7.2. On the Realization of Metrics by Surfaces with a Flat Point 7.3. Non-Bendable Surfaces With a Flattening 7.4. Bendable Surfaces with a Flattening 7.5. Surfaces of Revolution with Flattening at a Pole §8. Infinitesimal Bendings of Surfaces "in the small" 8.1. Equations of Infinitesimal Bendings 8.2. Rigidity "in the small" of Analytic Surfaces 8.3. Analytic Surfaces of Revolution with Flattening at a Pole 8.4. Rigid and Non-Bendable "in the large" Surfaces of Revolution 8.5. Non-Analytic Surfaces 8.6. Infinitesimal Bendings of the 2nd Order 8.7. Bendings of Troughs §9. Supplement. Bendings and Infinitesimal Bendings of Polyhedra 9.1. Introduction 9.2. Polyhedral Metrics and Their Isometric Immersions 9.3. Bendings of Polyhedra. Configuration Spaces of Polyhedra 9.4. Infinitesimal Bendings of Polyhedra and Their Connection with Bendings 9.5. Uniquely Determined Polyhedra 9.6. Non-Bendable Polyhedra 9.7. Bendable Polyhedra 9.8. Conjecture on the Invariance of the Volume of a Bendable Polyhedron §10. Concluding Remarks Comments on the References References Author Index Subject Index
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