The variational theory of geodesics 测地线的变分理论

Compact, self-contained text by noted theorist presents the most fundamental aspects of modern differential geometry as well as the basic tools required for the study of Morse theory. Advanced treatment; analytical rather than topological aspects of Morse theory emphasized. Discusses smooth manifolds, spaces of affine connection, Riemannian spaces, more. 1967 edition.

1. SMOOTH MANIFOLDS
1. Smooth Manifolds
2. Smooth Manifolds
3. Open Submanifolds. Property E
4. Vector Fields
5. Vector Fields on Coordinate Neighborhoods
6. Vectors
7. Linear Differential Forms
8. Covectors
9. Tensor Fields
10. Tensors. The Multiplications of Tensors and Tensor Fields
11. The Contraction of Tensors and Tensor Fields
12. Curves and Surfaces
13. The Extension of Tensor Fields
14. Submanifolds
15. Products of Manifolds
2. SPACES OF AFFINE CONNECTION
1. Affine Connections
2. The Curvature and Torsion Tensors
3. Covariant Differentiation Along a Curve
4. Parallel Translation Along a Curve
5. Covariant Differentiation of Tensor Fields
6. Geodesics
7. Normal Neighborhoods
8. Whitehead's Theorem
9. Differential Connection Forms
10. Cartan's Equations
3. RIEMANNIAN SPACES
1. Existence and Uniqueness of a Riemannian Connection
2. The Riemannian Curvature Tensor
3. Differential Connection Forms and the Metric Tensor
4. Arc Length
5. The Interior Metric
6. Minimizing Curves
7. Normal Convex Neighborhoods
8. A Lemma on Convergence
9. Complete Riemannian Spaces
10. Conditions for Completeness of Riemannian Spaces
4. THE VARIATIONAL PROPERTIES OF GEODESICS
1. Geodesics as Curves of Stationary Length
2. The Second Variation of Arc Length of a Geodesic
3. Jacobi Variations and dacobi Fields
4. Conjugate Points
5. Piecewise-Smooth and Discontinuous Vector Fields
6. Minimal Vector Fields
7. The Existence of Minimal Fields
8. Broken dacobi Fields
9. A Theorem on Isomorphism
10. Morse's Quadratic Form
11. Evaluation of the Index of a Point With the Aid of Morse's Form
12. Evaluation of the Index of an Interval With the Aid of Morse's Form
13. Bott's Quadratic Form. The Final Formulation of the Theorems on Indices.
APPENDIX FOCAL POINTS
1. The Second Quadratic Form of a Submanifold
2. Focal Points
3. Evaluation of the Index 入N
4. Proof of Inequality (4)
5. A REDUCTION THEOREM
1. Formulation of the Theorem
2. Remarks on the Formulation of the Reduction Theorem
3. Continuity of the Mappings a and/B
4. Completion of the Proof of the Reduction Theorem
5. A Generalized Reduction Theorem
6. Comparison of the Space ∩ With the Space ∩
INDEX