Introduction to Differential Geometry of Space Curves and Surfaces

Cover Page
 Title Page
 Copyright Page
 Preface
 Contents
 PART 1 THE THEORY OF CURVES AND SURFACES IN THREE-DIMENSIONAL EUCLIDEAN SPACE
 I. THE THEORY OF SPACE CURVES
 1. Introductory remarks about space curves
 2. Definitions
 3. Arc length
 4. Tangent, normal, and binormal
 5. Curvature and torsion of a curve given as the intersection of two surfaces
 6. Contact between curves and surfaces
 7. Tangent surface, involutes, and evolutes
 8. Intrinsic equations, fundamental existence theorem for space curves
 9. Helices
 Appendix I. 1. Existence theorem on linear differential equations. Miscellaneous Exercises III. THE METRIC: LOCAL INTRINSIC PROPERTIES OF A SURFACE
 1. Definition of a surface
 2. Curves on a surface
 3. Surfaces of revolution
 4. Helicoids
 5. Metric
 6. Direction coefficients
 7. Families of curves
 8. Isometric correspondence
 9. Intrinsic properties
 10. Geodesics
 11. Canonical geodesic equations
 12 Normal property of geodesics
 13. Existence theorems
 14. Geodesic parallels
 15. Geodesic curvature
 16. Gauss-Bonnet theorem
 17. Gaussian curvature
 18. Surfaces of constant curvature
 19. Conformal mapping
 20. Geodesic mapping. Appendix II. 1. The second existence theoremMiscellaneous Exercises II
 III. THE SECOND FUNDAMENTAL FORM: LOCAL NON-INTRINSIC PROPERTIES OF A SURFACE
 1. The second fundamental form
 2. Principal curvatures
 3. Lines of curvature
 4. Developables
 5. Developables associated with space curves
 6. Developables associated with curves on surfaces
 7. Minimal surfaces
 8. Ruled surfaces
 9. The fundamental equations of surface theory
 10. Parallel surfaces
 11. Fundamental existence theorem for surfaces
 Miscellaneous Exercises III
 IV. DIFFERENTIAL GEOMETRY OF SURFACES IN THE LARGE. 1. Introduction2. Compact surfaces whose points are umbilics
 3. Hilbert's lemma
 4. Compact surfaces of constant Gaussian or mean curvature
 5. Complete surfaces
 6. Characterization of complete surfaces
 7. Hilbert's theorem
 8. Conjugate points on geodesics
 9. Intrinsically defined surfaces
 10. Triangulation
 11. Two-dimensional Riemannian manifolds
 12. The problem of metrization
 13. The problem of continuation
 14. Problems of embedding and rigidity
 15. Conclusion
 PART 2 DIFFERENTIAL GEOMETRY OF n-DIMENSIONAL SPACE
 V. TENSOR ALGEBRA
 1. Vector spaces
 2. The dual space. 3. Tensor product of vector spaces4. Transformation formulae
 5. Contraction
 6. Special tensors
 7. Inner product
 8. Associated tensors
 9. Exterior algebra
 Miscellaneous Exercises V
 VI. TENSOR CALCULUS
 1. Differentiable manifolds
 2. Tangent vectors
 3. Affine tensors and tensorial forms
 4. Connexions
 5. Covariant differentiation
 6. Connexions over submanifolds
 7. Absolute derivation of tensorial forms.
 Appendix VI. 1. Tangent vectors to manifolds of class 
 Appendix VI. 2. Tensor-connexions
 Miscellaneous Exercises VI
 VII. RIEMANNIAN GEOMETRY
 1. Riemannian manifolds
 2. Metric.