Geometry and Topology (preliminary draft)
Differential geometry of curves and surfaces
Differentia geometry of curves and surfaces.
M. do Carmo: Prentice- Hall, 1976(25th printing)
Table of Contents
1. Curves: Parametrized Curves.
2. Regular Surfaces: Regular Surfaces; Inverse Images of Regular Values.
3. Geometry of the Gauss Map: Definition of the Gauss Map and Its Fundamental Properties.
4. Intrinsic Geometry of Surfaces: Isometrics; Conformal Maps.
5. Global Differential Geometry: Rigidity of the Sphere.
Differential Manifolds and Riemannian Geometry
An Introduction to Differentiable Manifolds and Riemannian Geometry.
W.M. Boothby: Academic Press, Inc., Orlando, FL, 1986
Contents
I. Introduction to Manifolds
1. Preliminary Comments on Rn
2. Rn and Euclidean Space 4
3. Topological Manifolds 6
4. Further Examples of Manifolds. Cutting andPasting Abstract Manifolds.
Some Examples 14
II. Functions of Several Variables and Mappings
1. Differentiability for Functions of Several Variables 20
2. Differentiability of Mappings and Jacobians 25
3. The Space of Tangent Vectors at a Point of Rn 29
4. Another Definition of Ta(Rn) 32
5. Vector Fields on Open Subsets of Rn
6. The Inverse Function Theorem 41
7. The Rank of a Mapping 46
III. Differentiable Manifolds and Submanifolds
1. The Definition of a Differentiable Manifold
2. Further Examples 59
3. Differentiable Functions and Mappings
4. Rank of a Mapping, Immersions 68
5. Submanifolds 74
6. Lie Groups 80 (*)
7. The Action of a Lie Group on a Manifold. Transformation Groups (*)
8. The Action of a Discrete Group on a Manifold 93 (*)
9. Covering Manifolds 98 (*)
IV. Vector Fields on a Manifold
1. The Tangent Space at a Point of a Manifold
2. Vector Field 113
3. One-Parameter and Local One-Parameter Groups Acting on a Manifold
4. The Existence Theorem for Ordinary Differential Equations 127
5. Some Examples of One-Parameter Groups Acting on a Manifold (*)
6. One-Parameter Subgroups of Lie Groups 142 (*)
7. The Lie Algebra of Vector Fields on a Manifold (*)
8. Frobenius's Theorem153 (*)
9. Homogeneous Spaces 160 (*)
V. Tensors and Tensor Fields on Manifolds
1. Tangent Covectors 171
Covectors on Manifolds 172
Covector Fields and Mappings 174
2. Bilinear Forms. The Riemannian Metric
3. Riemannian Manifolds as Metric Spaces
4. Partitions of Unity 186
Some Applications of the Partition of Unity
5. Tensor Fields192
Tensors on a Vector Space
Tensor Fields194
Mappings and Covariant Tensors 195
The Symmetrizing and Alternating Transformations
6. Multiplication of Tensors 199
Multiplication of Tensors on a Vector Space
Multiplication of Tensor Fields 201
Exterior Multiplication of Alternating Tensors
The Exterior Algebra on Manifolds 206
7. Orientation of Manifolds and the Volume Element
8. Exterior Differentiation212
An Application to Frobenius's Theorem 177
VI. Integration on Manifolds
1. Integration in Rn Domains of Integration 223
Basic Properties of the Riemann Integral 224
2. A Generalization to Manifolds229
Integration on Riemannian Manifolds
3. Integration on Lie Groups 237
4. Manifolds with Boundary 243
5. Stokes's Theorem for Manifolds 251
6. Homotopy of Mappings. The Fundamental Group 258 (*)
Homotopy of Paths and Loops. The Fundamental Group (*)
7. Some Applications of Differential Forms.
The de Rham Groups
The Homotopy Operator 268 (*)
8. Some Further Applications of de Rham Groups (*)
The de Rham Groups of Lie Groups 276 (*)
9. Covering Spaces and Fundamental Group 280 (*)
VII. Differentiation on Riemannian Manifolds
1. Differentiation of Vector Fields along Curves in Rn
The Geometry of Space Curves 292
Curvature of Plane Curves 296
2. Differentiation of Vector Fields on Submanifolds of Rn
Formulas for Covariant Derivatives 303
Differentiation of Vector Fields 305
3. Differentiation on Riemannian Manifolds 308
Constant Vector Fields and Parallel Displacement
4. Addenda to the Theory of Differentiation on a Manifold
The Curvature Tensor 316(*)
The Riemannian Connection and Exterior Differential Forms (*)
5. Geodesic Curves on Riemannian Manifolds 321(*)
6. The Tangent Bundle and Exponential Mapping. Normal Coordinates (*)
7. Some Further Properties of Geodesics 332(*)
8. Symmetric Riemannian Manifolds 340(*)
9. Some Examples 346 (*)
VIII. Curvature
1. The Geometry of Surfaces in E^3
The Principal Curvatures at a Point of a Surface 359
2. The Gaussian and Mean Curvatures of a Surface 363
The Theorema Egregium of Gauss 366
3. Basic Properties of the Riemann Curvature Tensor
4. Curvature Forms and the Equations of Structure
5. Differentiation of Covariant Tensor Fields 384(*)
6. Manifolds of Constant Curvature 391(*)
Spaces of Positive Curvature 394
Spaces of Zero Curvature 396
Spaces of Constant Negative Curvature
Other references:
M. Spivak, A comprehensive introduction to differential geometry
N. Hicks, Notes on differential geometry, Van Nostrand.
J. Milnor, Morse Theory
Basic topology
M. Armstrong, Springer, Undergraduate texts in mathematics
Table of contents:
Preface. 1: Introduction. 2: Continuity. 3: Compactness and connectedness. 4: Identification spaces. 5: The fundamental group. 6: Triangulations. 7: Surfaces. 8: Simplicial homology. 9: Degree and Lefschetz number. 10: Knots and covering spaces. Appendix: Generators and relations.
Algebraic topology
Algebraic topology
By A Hatcher (
Chapter 0. Some Underlying Geometric Notions
Homotopy and Homotopy Type. Cell Complexes. Operations on Spaces. Two Criteria for Homotopy Equivalence. The Homotopy Extension Property.
Chapter 1. Fundamental Group and Covering Spaces
1. Basic Constructions.
Paths and Homotopy. The Fundamental Group of the Circle. Induced Homomorphisms.
2. Van Kampen's Theorem
Free Products of Groups. The van Kampen Theorem. Applications to Cell Complexes.
3. Covering Spaces
Lifting Properties. The Classification of Covering Spaces. Deck Transformations and Group Actions.
4. Additional Topics (*)
Graphs and Free Groups. K(G,1) Spaces and Graphs of Groups.
Chapter 2. Homology
1. Simplicial and Singular Homology
Delta-Complexes. Simplicial Homology. Singular Homology. Homotopy Invariance. Exact Sequences and Excision. The Equivalence of Simplicial and Singular Homology.
2. Computations and Applications
Degree. Cellular Homology. Mayer-Vietoris Sequences. Homology with Coefficients.
3. The Formal Viewpoint (*)
Axioms for Homology. Categories and Functors.
4. Additional Topics (*)
Homology and Fundamental Group. Classical Applications. Simplicial Approximation.
Chapter 3. Cohomology
1. Cohomology Groups
The Universal Coefficient Theorem. Cohomology of Spaces.
2. Cup Product
The Cohomology Ring. A Kunneth Formula. Spaces with Polynomial Cohomology.
3. Poincare Duality
Orientations and Homology. The Duality Theorem. Cup Product and Duality. Other Forms of Duality.
4. Additional Topics (*)
The Universal Coefficient Theorem for Homology. The General Kunneth Formula. H-Spaces and Hopf Algebras. The Cohomology of SO(n). Bockstein Homomorphisms. Limits. More About Ext. Transfer Homomorphisms. Local Coefficients.
Chapter 4. Homotopy Theory (*)
1. Homotopy Groups
Definitions and Basic Constructions. Whitehead's Theorem. Cellular Approximation. CW Approximation.
2. Elementary Methods of Calculation
Excision for Homotopy Groups. The Hurewicz Theorem. Fiber Bundles. Stable Homotopy Groups.
3. Connections with Cohomology
The Homotopy Construction of Cohomology. Fibrations. Postnikov Towers. Obstruction Theory.
4. Additional Topics
Basepoints and Homotopy. The Hopf Invariant. Minimal Cell Structures. Cohomology of Fiber Bundles. The Brown Representability Theorem. Spectra and Homology Theories. Gluing Constructions. Eckmann-Hilton Duality. Stable Splittings of Spaces. The Loopspace of a Suspension. Symmetric Products and the Dold-Thom Theorem. Steenrod Squares and Powers.
Appendix
Topology of Cell Complexes. The Compact-Open Topology.
Other references:
J. Milnor, Topology from the differentiable viewpoint
R. Bott and L. Tu, Differential forms in algebraic topology
V. Guillemin, A. Pollack, Differential topology
Note: Chapters and sections with (*) shall not be covered in the contests.