Differential Geometry

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-Civita 77. Further properties of the displacement of Levi-Civita 78. A more general definition of absolute differentiation and displacement of Levi-Civita 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