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# Differential Geometry

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## Description

The first chapters of the book focus on the basic concepts and facts of analytic geometry, the theory of space curves, and the foundations of the theory of surfaces, including problems closely related to the first and second fundamental forms. The treatment of the theory of surfaces makes full use of the tensor calculus.

The later chapters address geodesics, mappings of surfaces, special surfaces, and the absolute differential calculus and the displacement of Levi-Cività. Problems at the end of each section (with solutions at the end of the book) will help students meaningfully review the material presented, and familiarize themselves with the manner of reasoning in differential geometry.

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## Product details

- Paperback | 384 pages
- 18 x 215 x 18.03mm | 394.63g
- 01 Jun 1991
- Dover Publications Inc.
- New York, United States
- English
- New edition
- New edition
- 0486667219
- 9780486667218
- 89,911

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## Table of contents

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

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