This volume studies smooth geometric structures using calculus. It develops curves, surfaces, manifolds, and curvature, forming the foundation for modern geometry and physics.
Part I. Curves and Surfaces
Chapter 1. Curves in Euclidean Space
1.1 Parametrized curves 1.2 Arc length 1.3 Curvature and torsion 1.4 Frenet frame 1.5 Examples
Chapter 2. Surfaces
2.1 Parametrized surfaces 2.2 Tangent planes 2.3 First fundamental form 2.4 Surface area 2.5 Examples
Chapter 3. Curvature of Surfaces
3.1 Second fundamental form 3.2 Gaussian curvature 3.3 Mean curvature 3.4 Principal curvatures 3.5 Examples
Part II. Manifolds
Chapter 4. Smooth Manifolds
4.1 Definitions 4.2 Charts and atlases 4.3 Smooth maps 4.4 Examples 4.5 Basic properties
Chapter 5. Tangent Spaces
5.1 Tangent vectors 5.2 Derivations 5.3 Tangent bundle 5.4 Vector fields 5.5 Examples
Chapter 6. Differential Forms
6.1 Exterior algebra 6.2 Differential forms 6.3 Exterior derivative 6.4 Integration on manifolds 6.5 Examples
Part III. Riemannian Geometry
Chapter 7. Riemannian Metrics
7.1 Inner products on tangent spaces 7.2 Length and distance 7.3 Geodesics 7.4 Examples 7.5 Applications
Chapter 8. Connections
8.1 Covariant derivative 8.2 Levi-Civita connection 8.3 Parallel transport 8.4 Examples 8.5 Applications
Chapter 9. Curvature
9.1 Riemann curvature tensor 9.2 Ricci curvature 9.3 Scalar curvature 9.4 Examples 9.5 Applications
Part IV. Global Differential Geometry
Chapter 10. Geodesic Structure
10.1 Completeness 10.2 Exponential map 10.3 Hopf–Rinow theorem 10.4 Applications 10.5 Examples
Chapter 11. Topology of Manifolds
11.1 Compactness 11.2 Orientation 11.3 Degree theory 11.4 Applications 11.5 Examples
Chapter 12. Gauss–Bonnet Theorem
12.1 Statement 12.2 Local and global curvature 12.3 Applications 12.4 Examples 12.5 Connections
Part V. Lie Groups and Symmetry
Chapter 13. Lie Groups as Manifolds
13.1 Smooth group structure 13.2 Examples 13.3 Lie algebras 13.4 Exponential map 13.5 Applications
Chapter 14. Group Actions
14.1 Actions on manifolds 14.2 Orbits and stabilizers 14.3 Quotient manifolds 14.4 Applications 14.5 Examples
Chapter 15. Homogeneous Spaces
15.1 Definitions 15.2 Examples 15.3 Geometric structures 15.4 Applications 15.5 Connections
Part VI. Advanced Topics
Chapter 16. Fiber Bundles
16.1 Definitions 16.2 Principal bundles 16.3 Associated bundles 16.4 Applications 16.5 Examples
Chapter 17. Connections and Curvature
17.1 Curvature forms 17.2 Gauge theory (overview) 17.3 Applications 17.4 Examples 17.5 Connections
Chapter 18. Symplectic Geometry (Overview)
18.1 Symplectic forms 18.2 Hamiltonian systems 18.3 Applications 18.4 Examples 18.5 Connections
Part VII. Applications
Chapter 19. Physics Applications
19.1 General relativity 19.2 Gauge theories 19.3 Classical mechanics 19.4 Applications 19.5 Examples
Chapter 20. Geometry and Topology
20.1 Interaction with topology 20.2 Geometric structures 20.3 Applications 20.4 Examples 20.5 Connections
Chapter 21. Computational Geometry
21.1 Discrete differential geometry 21.2 Numerical methods 21.3 Visualization 21.4 Applications 21.5 Examples
Part VIII. Research Directions
Chapter 22. Advanced Topics
22.1 Geometric analysis 22.2 Ricci flow 22.3 Minimal surfaces 22.4 Modern developments 22.5 Emerging areas
Chapter 23. Open Problems
23.1 Curvature classification 23.2 Geodesic behavior 23.3 Global structure questions 23.4 Computational challenges 23.5 Future directions
Chapter 24. Historical and Conceptual Notes
24.1 Development of differential geometry 24.2 Key contributors 24.3 Evolution of ideas 24.4 Cross-disciplinary impact 24.5 Summary
Appendix
A. Curvature formulas reference B. Standard manifolds C. Proof techniques checklist D. Tensor identities E. Cross-reference to other MSC branches