Book:T.J. Willmore/An Introduction to Differential Geometry

Subject Matter

 * Differential Geometry

Contents

 * Preface

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 theorems for space curves
 * 9. Helices
 * Appendix I. 1. Existence theorem on linear differential equations
 * Miscellaneous Exercises I


 * II. The Metric: Local Intinsic 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 theorem
 * Miscellaneous 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. Introduction
 * 2. 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 spaces
 * 4. 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 $$\infty$$
 * Appendix VI. 2. Tensor-connexions
 * Miscellaneous Exercises VI


 * VII. Riemannian Geometry
 * 1. Riemannian manifolds
 * 2. Metric
 * 3. The fundamental theorem of local Riemannian geometry
 * 4. Differential parameters
 * 5. Curvature tensors
 * 6. Geodesics
 * 7. Geodesic curvature
 * 8. Geometrical interpretation of the curvature tensor
 * 9. Special Riemannian spaces
 * 10. Parallel vectors
 * 11. Vector subspaces
 * 12. Parallel fields of planes
 * 13. Recurrent tensors
 * 14. Integrable distributions
 * 15. Riemann extensions
 * 16. É. Cartan's approach to Riemannian geometry
 * 17. Euclidean tangent metrics
 * 18. Euclidean osculating metrics
 * 19. The equations of structure
 * 20. Global Riemannian geometry
 * Bibliographies on harmonic spaces, recurrent spaces, parallel distributions, Riemann extensions
 * Miscellaneous Exercises VII


 * VIII. Applications of Tensor Methods to Surface Theory
 * 1. The Serret-Frenet formula
 * 2. The induced metric
 * 3. The fundamental formulae of surface theory
 * 4. Normal curvature and geodesic torsion
 * 5. The method of moving frames
 * Miscellaneous Exercises VIII


 * Exercises


 * Suggestions for Further Reading


 * Index