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

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T.J. Willmore: An Introduction to Differential Geometry

Published $1959$, Oxford University Press

ISBN 0 19 561110 1.

Subject Matter


Preface (Liverpool, 1958)
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

Suggestions for Further Reading