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.
Author(s): Victor V. Prasolov
Series: Moscow Lectures 8
Edition: 1
Publisher: Springer Nature Switzerland AG
Year: 2022
Language: English
Pages: 271
City: Cham, Switzerland
Tags: Differential Geometry, Curves, Surfaces, Riemannian Manifolds, Lie Groups
Preface
Basic Notation
Contents
1 Curves in the Plane
1.1 Curvature and the Frenet–Serret Formulas
1.2 Osculating Circles
1.3 The Total Curvature of a Closed Plane Curve
1.4 Four-Vertex Theorem
1.5 The Natural Equation of a Plane Curve
1.6 Whitney–Graustein Theorem
1.7 Tube Area and Steiner's Formula
1.8 The Envelope of a Family of Curves
1.9 Evolute and Involute
1.10 Isoperimetric Inequality
1.11 Affine Unimodular Differential Geometry
1.12 Projective Differential Geometry
1.13 The Measure of the Set of Lines Intersecting a Given Curve
1.14 Solutions of Problems
2 Curves in Space
2.1 Curvature and Torsion: The Frenet–Serret Formulas
2.2 An Osculating Plane
2.3 Total Curvature of a Closed Curve
2.4 Bertrand Curves
2.5 The Frenet–Serret Formulas in Many-Dimensional Space
2.6 Solutions of Problems
3 Surfaces in Space
3.1 The First Quadratic Form
3.2 The Darboux Frame of a Curve on a Surface
3.3 Geodesics
3.4 The Second Quadratic Form
3.5 Gaussian Curvature
3.6 Gaussian Curvature and Differential Forms
3.7 The Gauss–Bonnet Theorem
3.8 Christoffel Symbols
3.9 The Spherical Gauss Map
3.10 The Geodesic Equation
3.11 Parallel Transport Along a Curve
3.12 Covariant Differentiation
3.13 The Gauss and Codazzi–Mainardi Equations
3.14 Riemann Curvature Tensor
3.15 Exponential Map
3.16 Lines of Curvature and Asymptotic Lines
3.17 Minimal Surfaces
3.18 The First Variation Formula
3.19 The Second Variation Formula
3.20 Jacobi Vector Fields and Conjugate Points
3.21 Jacobi's Theorem on a Normal Spherical Image
3.22 Surfaces of Constant Gaussian Curvature
3.23 Rigidity (Unbendability) of the Sphere
3.24 Convex Surfaces: Hadamard's Theorem
3.25 The Laplace–Beltrami Operator
3.26 Solutions of Problems
4 Hypersurfaces in Rn+1: Connections
4.1 The Weingarten Operator
4.2 Connections on Hypersurfaces
4.3 Geodesics on Hypersurfaces
4.4 Convex Hypersurfaces
4.5 Minimal Hypersurfaces
4.6 Steiner's Formula
4.7 Connections on Vector Bundles
4.8 Geodesics
4.9 The Curvature Tensor and the Torsion Tensor
4.10 The Curvature Matrix of a Connection
4.11 Solutions of Problems
5 Riemannian Manifolds
5.1 Levi-Cività Connection
5.2 Symmetries of the Riemann Tensor
5.3 Geodesics on Riemannian Manifolds
5.4 The Hopf–Rinow Theorem
5.5 The Existence of Complete Riemannian Metrics
5.6 Covariant Differentiation of Tensors
5.7 Sectional Curvature
5.8 Ricci Tensor
5.9 Riemannian Submanifolds
5.10 Totally Geodesic Submanifolds
5.11 Jacobi Fields and Conjugate Points
5.12 Product of Riemannian Manifolds
5.13 Holonomy
5.14 Commutator and Curvature
5.15 Solutions of Problems
6 Lie Groups
6.1 Lie Groups and Algebras
6.2 Adjoint Representation and the Killing Form
6.3 Connections and Metrics on Lie Groups
6.4 Maurer–Cartan Equations
6.5 Invariant Integration on a Compact Lie Group
6.6 Lie Derivative
6.7 Infinitesimal Isometries
6.8 Homogeneous Spaces
6.9 Symmetric Spaces
6.10 Solutions of Problems
7 Comparison Theorems, Curvature and Topology, and Laplacian
7.1 The Simplest Comparison Theorems
7.2 The Cartan–Hadamard Theorem
7.3 Manifolds of Positive Curvature
7.4 Manifolds of Constant Curvature
7.5 Laplace Operator
7.6 Solutions of Problems
8 Appendix
8.1 Differentiation of Determinants
8.2 Jacobi Identity for the Commutator of Vector Fields
8.3 The Differential of a 1-Form
Bibliography
Index
