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Differential Geometry 2022 ed. [Hardback]

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  • Formāts: Hardback, 271 pages, height x width: 235x155 mm, weight: 594 g, 1 Illustrations, black and white; XI, 271 p. 1 illus., 1 Hardback
  • Sērija : Moscow Lectures 8
  • Izdošanas datums: 11-Feb-2022
  • Izdevniecība: Springer Nature Switzerland AG
  • ISBN-10: 3030922480
  • ISBN-13: 9783030922481
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Differential Geometry 2022 ed.Palielināt
  • Formāts: Hardback, 271 pages, height x width: 235x155 mm, weight: 594 g, 1 Illustrations, black and white; XI, 271 p. 1 illus., 1 Hardback
  • Sērija : Moscow Lectures 8
  • Izdošanas datums: 11-Feb-2022
  • Izdevniecība: Springer Nature Switzerland AG
  • ISBN-10: 3030922480
  • ISBN-13: 9783030922481
Citas grāmatas par šo tēmu:
Permanent link: https://www.kriso.lv/db/9783030922481.html
Keywords:
This book combines the classical and contemporary approaches to differential geometry. An introduction to the Riemannian geometry of manifolds is preceded by a detailed discussion of properties of curves and surfaces.

The chapter on the differential geometry of plane curves considers local and global properties of curves, evolutes and involutes, and affine and projective differential geometry. Various approaches to Gaussian curvature for surfaces are discussed. The curvature tensor, conjugate points, and the Laplace-Beltrami operator are first considered in detail for two-dimensional surfaces, which facilitates studying them in the many-dimensional case. A separate chapter is devoted to the differential geometry of Lie groups.


Recenzijas

All chapters are supplemented with solutions of the problems scattered throughout the text. Designed as a text for a lecturer course on the subject, it is perfect and can be recommended for students interested in this classical field. (Ivailo. M. Mladenov, zbMATH 1498.53001, 2022)

1 Curves in the Plane
1(46)
1.1 Curvature and the Frenet-Serret Formulas
3(3)
1.2 Osculating Circles
6(1)
1.3 The Total Curvature of a Closed Plane Curve
7(3)
1.4 Four-Vertex Theorem
10(3)
1.5 The Natural Equation of a Plane Curve
13(1)
1.6 Whitney-Graustein Theorem
14(1)
1.7 Tube Area and Steiner's Formula
15(1)
1.8 The Envelope of a Family of Curves
16(5)
1.9 Evolute and Involute
21(2)
1.10 Isoperimetric Inequality
23(3)
1.11 Affine Unimodular Differential Geometry
26(3)
1.12 Projective Differential Geometry
29(4)
1.13 The Measure of the Set of Lines Intersecting a Given Curve
33(3)
1.14 Solutions of Problems
36(11)
2 Curves in Space
47(18)
2.1 Curvature and Torsion: The Frenet-Serret Formulas
47(4)
2.2 An Osculating Plane
51(2)
2.3 Total Curvature of a Closed Curve
53(2)
2.4 Bertrand Curves
55(1)
2.5 The Frenet-Serret Formulas in Many-Dimensional Space
56(1)
2.6 Solutions of Problems
57(8)
3 Surfaces in Space
65(80)
3.1 The First Quadratic Form
66(2)
3.2 The Darboux Frame of a Curve on a Surface
68(2)
3.3 Geodesies
70(2)
3.4 The Second Quadratic Form
72(3)
3.5 Gaussian Curvature
75(2)
3.6 Gaussian Curvature and Differential Forms
77(3)
3.7 The Gauss-Bonnet Theorem
80(4)
3.8 Christoffel Symbols
84(3)
3.9 The Spherical Gauss Map
87(1)
3.10 The Geodesic Equation
88(1)
3.11 Parallel Transport Along a Curve
89(3)
3.12 Covariant Differentiation
92(5)
3.13 The Gauss and Codazzi-Mainardi Equations
97(2)
3.14 Riemann Curvature Tensor
99(1)
3.15 Exponential Map
100(4)
3.16 Lines of Curvature and Asymptotic Lines
104(4)
3.17 Minimal Surfaces
108(3)
3.18 The First Variation Formula
111(2)
3.19 The Second Variation Formula
113(3)
3.20 Jacobi Vector Fields and Conjugate Points
116(6)
3.21 Jacobi's Theorem on a Normal Spherical Image
122(2)
3.22 Surfaces of Constant Gaussian Curvature
124(3)
3.23 Rigidity (Unbendability) of the Sphere
127(2)
3.24 Convex Surfaces: Hadamard's Theorem
129(1)
3.25 The Laplace-Beltrami Operator
129(5)
3.26 Solutions of Problems
134(11)
4 Hypersurfaces in Rn+1: Connections
145(24)
4.1 The Weingarten Operator
145(3)
4.2 Connections on Hypersurfaces
148(1)
4.3 Geodesies on Hypersurfaces
149(1)
4.4 Convex Hypersurfaces
149(1)
4.5 Minimal Hypersurfaces
150(2)
4.6 Steiner's Formula
152(1)
4.7 Connections on Vector Bundles
153(3)
4.8 Geodesies
156(3)
4.9 The Curvature Tensor and the Torsion Tensor
159(4)
4.10 The Curvature Matrix of a Connection
163(4)
4.11 Solutions of Problems
167(2)
5 Riemannian Manifolds
169(42)
5.1 Levi-Civita Connection
169(2)
5.2 Symmetries of the Riemann Tensor
171(2)
5.3 Geodesies on Riemannian Manifolds
173(2)
5.4 The Hopf-Rinow Theorem
175(3)
5.5 The Existence of Complete Riemannian Metrics
178(2)
5.6 Covariant Differentiation of Tensors
180(3)
5.7 Sectional Curvature
183(3)
5.8 Ricci Tensor
186(1)
5.9 Riemannian Submanifolds
187(4)
5.10 Totally Geodesic Submanifolds
191(1)
5.11 Jacobi Fields and Conjugate Points
192(5)
5.12 Product of Riemannian Manifolds
197(2)
5.13 Holonomy
199(1)
5.14 Commutator and Curvature
200(5)
5.15 Solutions of Problems
205(6)
6 Lie Groups
211(34)
6.1 Lie Groups and Algebras
211(7)
6.2 Adjoint Representation and the Killing Form
218(2)
6.3 Connections and Metrics on Lie Groups
220(3)
6.4 Maurer-Cartan Equations
223(2)
6.5 Invariant Integration on a Compact Lie Group
225(3)
6.6 Lie Derivative
228(4)
6.7 Infinitesimal Isometries
232(3)
6.8 Homogeneous Spaces
235(4)
6.9 Symmetric Spaces
239(2)
6.10 Solutions of Problems
241(4)
7 Comparison Theorems, Curvature and Topology, and Laplacian
245(12)
7.1 The Simplest Comparison Theorems
245(2)
7.2 The Cartan-Hadamard Theorem
247(2)
7.3 Manifolds of Positive Curvature
249(2)
7.4 Manifolds of Constant Curvature
251(2)
7.5 Laplace Operator
253(2)
7.6 Solutions of Problems
255(2)
8 Appendix
257(4)
8.1 Differentiation of Determinants
257(1)
8.2 Jacobi Identity for the Commutator of Vector Fields
258(1)
8.3 The Differential of a 1 - Form
259(2)
Bibliography 261(4)
Index 265
Victor Prasolov, born 1956, is a permanent teacher of mathematics at the Independent University of Moscow. He published two books with Springer, Polynomials and Algebraic Curves. Towards Moduli Spaces (jointly with M. E. Kazaryan and S. K. Lando) and eight books with AMS, including Problems and Theorems in Linear Algebra, Intuitive Topology, Knots, Links, Braids, and 3-Manifolds (jointly with A. B. Sossinsky), and Elliptic Functions and Elliptic Integrals (jointly with Yu. Solovyev).