Differential Geometry

An introductory textbook on the differential geometry of curves and surfaces in three-dimensional Euclidean space, presented in its simplest, most essential form, but with many explanatory details, figures and examples, and in a manner that conveys the theoretical and practical importance of the different concepts, methods and results involved. With problems at the end of each section, and solutions listed at the end of the book. 99 illustrations.

PREFACE
CHAPTER I. PRELIMINARIES
1. Notation
2. Nature and purpose of differential geometry
3. Concept of mapping. Coordinates in Euclidean space
4. Vectors in Euclidean space
5. Basic rules of vector calculus in Euclidean space
CHAPTER II. THEORY OF CURVES
6. The concept of a curve in differential geometry
7. Further remarks on the concept of a curve
8. Examples of special curves
9. Arc length
10. Tangent and normal plane
11. Osculating plane
12. "Principal normal, curvature, osculating circle "
13. Binormal. Moving trihedron of a curve
14. Torsion
15. Formulae of Frenet
16. "Motion of the trihedron, vector of Darboux "
17. Spherical images of a curve
18. Shape of a curve in the neighbourhood of any of its points (canonical representation)
19. "Contact, osculating sphere "
20. Natural equations of a curve
21. Examples of curves and their natural equations
22. Involutes and evolutes
23. Bertrand curves
CHAPTER III. CONCEPT OF A SURFACE. FIRST FUNDAMENTAL FORM. FOUNDATIONS OF TENSOR.CALCULUS
24. Concept of a surface in differential geometry
25. "Further remarks on the representation of surfaces, examples "
26. "Curves on a surface, tangent plane to a surface "
27. First fundamental form. Concept of Riemannian geometry. Summation convention
28. Properties of the first fundamental form
29. Contravariant and covariant vectors
30. "Contravariant, covariant, and mixed tensors "
31. Basic rules of tensor calculus
32. Vactors in a surface. The contravariant metric tensor
33. Special tensors
34. Normal to a surface
35. Measurement of lengths and angles in a surface
36. Area
37. Remarks on the definition of area
CHAPTER IV. SECOND FUNDAMENTAL FORM. GAUSSIAN AND MEAN CURVATURE OF A SURFACE
38. Second fundamental form
39. Arbitrary and nonnal sections of a surface. Meusnier's theorem. Asymptotic lines
40. "Elliptic, parabolic, and hyperbolic points of a surface "
41. Principal curvature. Lines of curvature. Gaussian and mean curvature
42. Euler's theorem. Dupin's indicatrix
43. Torus
44. Flat points. Saddle points of higher type
45. Formulae of Weingarten and Gauss
46. Integrability conditions of the formulae of Weingarten and Gauss. Curvature tensors. Theorema. egregium
47. Properties of the Christoffel symbols
48. Umbilics
CHAPTER V. GEODESIC CURVATURE AND GEODESICS
49. Geodesic curvature
50. Geodesics
51. Arcs of minimum length
52. Geodesic parallel coordinates
53. Geodesic polar coordinates
54. Theorem of Gauss-Bonnet. Integral curvature
55. Application of the Gauss-Bonnet theorem to closed surfaces
CHAPTER VI. MAPPINGS
56. Preliminaries
57. Isometric mapping. Bending. Concept of intrinsic geometry of a surface
58. "Ruled surfaces, developable surfaces "
59. Spherical image of a surface. Third fundamental form. Isometric mapping of developable surfaces
60. Conjugate directions. Conjugate families of curves. Developable surfaces contacting a surface.
61. Conformal mapping
62. Conformal mnpping of surfaces into a plane
63. Isotropic curves and isothermic coordinates
64. The Bergman metric
65. Conformal mapping of a sphere into a plane. Stereographic and Mercator projection
66. Equiareal mappings
67. "Equiareal mapping of spheres into planes. Mappings of Lambert, Sanson, and Bonne "
68. Conformal mapping of the Euclidean space
CHAPTER VII. ABSOLUTE DIFFERENTIATION AND PARALLEL DISPLACEMENT
69. Concept of absolute differentiation
70. Absolute differentiation of tensors of first order
71. Absolute differentiation of tensors of arbitrary order
72. Further properties of absolute differentiation
73. Interchange of the order of absolute differentiation. The Ricci identity
74. Bianchi identities
75. Differential parameters of Beltrami
76. Definition of the displacement of Levi-Cività
77. Further properties of the displacement of Levi-Cività
78. A more general definition of absolute differentiation and displacement of Levi-Cività
CHAPTER VIII. SPECIAL SURFACES
79. Definition and simple properties of minimal surfaces
80. Surfaces of smallest area
81. Examples of minimal surfaces
82. Relations between function theory and minimal surfaces. The formulae of Weierstrass
83. Minimal surfaces as translation surfaces with isotropic generators
84. Modular surfaces of analytic functions
85. Envelope of a one-parameter family of surfaces
86. Developable surfaces as envelopes of families of planes
87. "Envelope of the osculating, normal, and rectifying planes of a curve, polar surface "
88. Centre surfaces of a surface
89. Parallel surfaces
90. Surfaces of constant Gaussian curvature
91. Isometric mapping of surfaces of constant Gaussian curvature
92. Spherical surfaces of revolution
93. Pseudospherical surfaces of revolution
94. Goodesic mapping
95. Geodesic mapping of surfaces of constant Gaussian curvature
96. Surfaces of constant Gaussian curvature and non-Euclidean geometry
ANSWERS TO PROBLEMS
COLLECTION OF FORMULAE
BIBLIOGRAPHY
INDEX