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

T.J. Willmore: An Introduction to Differential Geometry

Published $\text {1959}$, Oxford University Press

ISBN 0 19 561110 1

Subject Matter

• Differential Geometry

Contents

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

Exercises

Suggestions for Further Reading

Index

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