| 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 |
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