This textbook serves as an introduction to modern differential geometry at a level accessible to advanced undergraduate and master's students. It places special emphasis on motivation and understanding, while developing a solid intuition for the more abstract concepts. In contrast to graduate level references, the text relies on a minimal set of prerequisites: a solid grounding in linear algebra and multivariable calculus, and ideally a course on ordinary differential equations. Manifolds are introduced intrinsically in terms of coordinate patches glued by transition functions. The theory is presented as a natural continuation of multivariable calculus; the role of point-set topology is kept to a minimum.
Questions sprinkled throughout the text engage students in active learning, and encourage classroom participation. Answers to these questions are provided at the end of the book, thus making it ideal for independent study. Material is further reinforced with homework problems ranging from straightforward to challenging.
Author(s): Gal Gross, Eckhard Meinrenken
Series: Springer Undergraduate Mathematics Series
Publisher: Springer
Year: 2023
Language: English
Pages: 347
City: Cham
Tags: Manifolds, Vector Fields, Differential Forms, Vector Bundles
Preface
Contents
1 Introduction
1.1 A Very Short History
1.2 The Concept of Manifolds: Informal Discussion
1.3 Manifolds in Euclidean Space
1.4 Intrinsic Descriptions of Manifolds
1.5 Soccer Balls and Linkages
1.6 Surfaces
1.7 Problems
2 Manifolds
2.1 Atlases and Charts
2.2 Definition of Manifold
2.3 Examples of Manifolds
2.3.1 Spheres
2.3.2 Real Projective Spaces
2.3.3 Complex Projective Spaces*
2.3.4 Real Grassmannians*
2.3.5 Complex Grassmannians*
2.4 Open Subsets
2.5 Compactness
2.6 Orientability
2.7 Building New Manifolds
2.7.1 Disjoint Union
2.7.2 Products
2.7.3 Connected Sums*
2.7.4 Quotients*
2.8 Problems
3 Smooth Maps
3.1 Smooth Functions on Manifolds
3.2 The Hausdorff Property via Smooth Functions
3.3 Smooth Maps Between Manifolds
3.4 Composition of Smooth Maps
3.5 Diffeomorphisms of Manifolds
3.6 Examples of Smooth Maps
3.6.1 Products, Diagonal Maps
3.6.2 The Diffeomorphisms RP1.5-.5.5-.5.5-.5.5-.5S1 and CP1.5-.5.5-.5.5-.5.5-.5S2*
3.6.3 Maps to and from Projective Space*
3.7 The Hopf Fibration*
3.8 Problems
4 Submanifolds
4.1 Submanifolds
4.2 The Rank of a Smooth Map
4.2.1 The Rank of the Jacobian Matrix
4.2.2 The Rank of Smooth Maps Between Manifolds
4.3 Smooth Maps of Maximal Rank
4.3.1 Local Diffeomorphisms
4.3.2 Submersions
4.3.3 Example: The Steiner Surface*
4.3.4 Quotient Maps*
4.3.5 Immersions
4.3.6 Further Remarks on Embeddings and Immersions
4.4 Problems
5 Tangent Spaces
5.1 Intrinsic Definition of Tangent Spaces
5.2 Tangent Maps
5.2.1 Definition of the Tangent Map, Basic Properties
5.2.2 Coordinate Description of the Tangent Map
5.2.3 Tangent Spaces of Submanifolds
5.2.4 Example: Steiner's Surface Revisited*
5.3 Problems
6 Vector Fields
6.1 Vector Fields as Derivations
6.2 Lie Brackets
6.3 Related Vector Fields*
6.4 Flows of Vector Fields
6.4.1 Solution Curves
6.4.2 Existence and Uniqueness for Open Subsets of Rm
6.4.3 Existence and Uniqueness for Vector Fields on Manifolds
6.4.4 Flows
6.4.5 Complete Vector Fields
6.5 Geometric Interpretation of the Lie Bracket
6.6 Frobenius Theorem
6.7 Problems
7 Differential Forms
7.1 Review: Differential Forms on Rm
7.2 Dual Spaces
7.3 Cotangent Spaces
7.4 1-forms
7.5 Pullbacks of Function and 1-forms
7.6 Integration of 1-forms
7.7 k-forms
7.7.1 2-forms
7.7.2 k-forms
7.7.3 Wedge Product
7.7.4 Exterior Differential
7.8 Lie Derivatives and Contractions*
7.9 Pullbacks
7.10 Problems
8 Integration
8.1 Integration of Differential Forms
8.1.1 Integration Over Open Subsets of Rm
8.1.2 Integration Over Manifolds
8.1.3 Integration Over Oriented Submanifolds
8.2 Stokes' Theorem
8.3 Winding Numbers and Mapping Degrees
8.3.1 Invariance of Integrals
8.3.2 Winding Numbers
8.3.3 Mapping Degree
8.4 Volume Forms
8.5 Applications to Differential Geometry of Surfaces
8.5.1 Euler Characteristic of Surfaces
8.5.2 Rotation Numbers for Vector Fields
Index of a Vector Field
Rotation Numbers Along Embedded Circles
8.5.3 Poincaré Theorem
8.5.4 Gauss-Bonnet Theorem
8.6 Problems
9 Vector Bundles
9.1 The Tangent Bundle
9.2 Vector Fields Revisited
9.3 The Cotangent Bundle
9.4 Vector Bundles
9.5 Some Constructions with Vector Bundles
9.6 Sections of Vector Bundles
9.7 Problems
Notions from Set Theory
A.1 Countability
A.2 Equivalence Relations
Notions from Algebra
B.1 Permutations
B.2 Algebras
B.2.1 Definition and Examples
B.2.2 Homomorphisms of Algebras
B.2.3 Derivations of Algebras
B.2.4 Modules over Algebras
B.3 Dual Spaces and Quotient Spaces
Topological Properties of Manifolds
C.1 Topological Spaces
C.2 Manifolds Are Second Countable
C.3 Manifolds Are Paracompact
C.4 Partitions of Unity
Hints and Answers to In-text Questions
References
List of Symbols
Index
