Elementary Differential Geometry

Elementary Differential Geometry focuses on the elementary account of the geometry of curves and surfaces.

The book first offers information on calculus on Euclidean space and frame fields. Topics include structural equations, connection forms, frame fields, covariant derivatives, Frenet formulas, curves, mappings, tangent vectors, and differential forms. The publication then examines Euclidean geometry and calculus on a surface. Discussions focus on topological properties of surfaces, differential forms on a surface, integration of forms, differentiable functions and tangent vectors, congruence of curves, derivative map of an isometry, and Euclidean geometry.

The manuscript takes a look at shape operators, geometry of surfaces in E, and Riemannian geometry. Concerns include geometric surfaces, covariant derivative, curvature and conjugate points, Gauss-Bonnet theorem, fundamental equations, global theorems, isometries and local isometries, orthogonal coordinates, and integration and orientation.

The text is a valuable reference for students interested in elementary differential geometry.

Auteur

Barrett O'Neill is currently a Professor in the Department of Mathematics at the University of California, Los Angeles. He has written two other books in advanced mathematics.

Contenu

Preface
Introduction
Chapter I. Calculus on Euclidean Space

1. Euclidean Space
2. Tangent Vectors
3. Directional Derivatives
4. Curves in E3
5. 1-Forms
6. Differential Forms
7. Mappings
8. Summary
   Chapter II. Frame Fields
9. Dot Product
10. Curves
11. The Frenet Formulas
12. Arbitrary-Speed Curves
13. Covariant Derivatives
14. Frame Fields
15. Connection Forms
16. The Structural Equations
17. Summary
    Chapter III. Euclidean Geometry
18. Isometries of E3
19. The Derivative Map of an Isometry
20. Orientation
21. Euclidean Geometry
22. Congruence of Curves
23. Summary
    Chapter IV. Calculus on a Surface
24. Surfaces in E3
25. Patch Computations
26. Differentiable Functions and Tangent Vectors
27. Differential Forms on a Surface
28. Mappings of Surfaces
29. Integration of Forms
30. Topological Properties of Surfaces
31. Manifolds
32. Summary
    Chapter V. Shape Operators
33. The Shape Operator of M C E3
34. Normal Curvature
35. Gaussian Curvature
36. Computational Techniques
37. Special Curves in a Surface
38. Surfaces of Revolution
39. Summary
    Chapter VI. Geometry of Surfaces in E3
40. The Fundamental Equations
41. Form Computations
42. Some Global Theorems
43. Isometries and Local Isometries
44. Intrinsic Geometry of Surfaces in E3
45. Orthogonal Coordinates
46. Integration and Orientation
47. Congruence of Surfaces
48. Summary
    Chapter VII. Riemannian Geometry
49. Geometric Surfaces
50. Gaussian Curvature
51. Covariant Derivative
52. Geodesics
53. Length-Minimizing Properties of Geodesics
54. Curvature and Conjugate Points
55. Mappings that Preserve Inner Products
56. The Gauss-Bonnet Theorem
57. Summary
    Bibliography
    Answer to Odd-Numbered Exercises
    Index