Differential Geometry

Differential Geometry......Page 3
CONTENTS......Page 17
1. The vector notation......Page 23
2. Addition of vectors......Page 24
5. Scalar product......Page 25
6. Vector product......Page 27
7. Scalar triple product......Page 28
8. Invariance under orthogonal transformations......Page 29
9. Vector calculus......Page 30
2. Regular curves......Page 34
3. Change of parameters......Page 36
5. Tangent lines and tangent vectors of a curve......Page 38
6. Orientation of a curve......Page 40
7. Length of a curve......Page 41
8. Arc length as an invariant......Page 42
9. Curvature of plane curves......Page 43
10. The normal vector and the sign of k......Page 45
11. Formulas for κ......Page 48
12. Existence of a plane curve for given curvature κ......Page 49
13. Frenet equations for plane curves......Page 50
14. Evolute and involute of a plane curve......Page 51
15. Envelopes of families of curves......Page 53
16. The Jordan theorem as a problem in differential geometry in the large......Page 56
17. Additional properties of Jordan curves......Page 63
18. The total curvature of a regular Jordan curve......Page 67
19. Simple closed curves with κ ≠ 0 as boundaries of convex point sets......Page 68
20. Four vertex theorem......Page 70
1. Regular curves......Page 75
3. Curvature of space curves......Page 76
4. Principal normal and osculating plane......Page 77
6. Torsion τ of a space curve......Page 79
8. Rigid body motions and the rotation vector......Page 80
9. The Darboux vector......Page 84
11. The sign of τ......Page 85
12. Canonical representation of a curve......Page 86
13. Existence and uniqueness of a space curve for given κ (S), τ (S)......Page 87
14. What about κ = 0?......Page 89
15. Another way to define space curves......Page 90
16. Some special curves......Page 92
1. Regular surfaces in Euclidean space......Page 96
2. Change of parameters......Page 97
3. Curvilinear coordinate curves on a surface......Page 98
4. Tangent plane and normal vector......Page 99
6. Invariance of the first fundamental form......Page 100
7. Angle measurement on surfaces......Page 102
8. Area of a surface......Page 104
9. A few examples......Page 105
10. Second fundamental form of a surface......Page 107
11. Osculating paraboloid......Page 108
12. Curvature of curves on a surface......Page 110
13. Principal directions and principal curvatures......Page 113
14. Mean curvature H and Gaussian curvature K......Page 114
15. Another definition of the Gaussian curvature K......Page 115
16. Lines of curvature......Page 117
17. Third fundamental form......Page 120
18. Characterization of the sphere as a locus of umbilical points......Page 121
20. Torsion of asymptotic lines......Page 122
21. Introduction of special parameter curves......Page 123
23. Embedding a given arc in a system of parameter curves......Page 125
24. Analogues of polar coordinates on a surface......Page 126
1. Surfaces of revolution......Page 131
2. Developable surfaces in the small made up of parabolic points......Page 136
3. Edge of regression of a developable......Page 140
4. Why the name developable?......Page 142
5. Developable surfaces in the large1......Page 143
6. Developables as envelopes of planes......Page 151
1. Introduction......Page 155
2. The Gauss equations......Page 156
3. The Christoffel symbols evaluated......Page 157
5. Some observations about the partial differential equations......Page 158
6. Uniqueness of a surface for given gik and Lik......Page 160
7. The theorema egregium of Gauss......Page 161
8. How Gauss may have hit upon his theorem......Page 163
9. Compatibility conditions in general......Page 165
11. The Gauss theorema egregium again......Page 166
12. Existence of a surface with given gik and Lik......Page 168
13. An application of the general theory to a problem in the large......Page 170
1. Introduction. Motivations for the basic concepts......Page 173
2. Approximate local parallelism of vectors in a surface......Page 177
3. Parallel transport of vectors along curves in the sense of Levi-Civita......Page 179
4. Properties of parallel fields of vectors along curves......Page 182
5. Parallel transport is independent of the path only for surfaces having K = 0......Page 184
6. The curvature of curves in a surface : the geodetic curvature......Page 185
7. First definition of geodetic lines: lines with κg = 0......Page 187
8. Geodetic lines as candidates for shortest arcs......Page 189
9. Straight lines as shortest arcs in the Euclidean plane......Page 190
10. A general necessary condition for a shortest arc......Page 193
11. Geodesies in the small and geodetic coordinate systems......Page 196
12. Geodesies as shortest arcs in the small......Page 200
13. Further developments relating to geodetic coordinate systems......Page 201
14. Surfaces of constant Gaussian curvature......Page 205
15. Parallel fields from a new point of view......Page 206
16. Models provided by differential geometry for non-Euclidean geometries......Page 207
17. Parallel transport of a vector around a simple closed curve......Page 213
18. Derivation of the Gauss-Bonnet formula......Page 217
19. Consequences of the Gauss-Bonnet formula......Page 218
20. Tchebychef nets......Page 220
1. Introduction. Definition of n-dimensional manifolds......Page 225
2. Definition of a Riemannian manifold......Page 228
3. Facts from topology relating to two-dimensional manifolds......Page 233
4. Surfaces in three-dimensional space......Page 239
5. Abstract surfaces as metric spaces......Page 240
6. Complete surfaces and the existence of shortest arcs......Page 242
7. Angle comparison theorems for geodetic triangles......Page 248
8. Geodetically convex domains......Page 253
9. The Gauss-Bonnet formula applied to closed surfaces......Page 259
10. Vector fields on surfaces and their singularities......Page 261
11. Poincaré's theorem on the sum of the indices on closed surfaces......Page 266
12. Conjugate points. Jacobi's conditions for shortest arcs......Page 269
13. The theorem of Bonnet-Hopf-Rinow......Page 276
14. Synge's theorem in two dimensions......Page 277
15. Covering surfaces of complete surfaces having K ≤ 0......Page 281
16. Hilbert's theorem on surfaces in E3 with K ≡–1......Page 287
17. The form of complete surfaces of positive curvature in three-dimensional space......Page 294
1. Introduction......Page 304
2. Affine geometry in curvilinear coordinates......Page 306
3. Tensor calculus in Euclidean spaces......Page 309
4. Tensor calculus in mechanics and physics......Page 314
5. Tensors in a Riemannian space......Page 316
6. Basic concepts of Riemannian geometry......Page 318
7. Parallel displacement. Necessary condition for Euclidean metrics......Page 322
8. Normal coordinates. Curvature in Riemannian geometry......Page 329
9. Geodetic lines as shortest connections in the small......Page 332
10. Geodetic lines as shortest connections in the large......Page 333
11. Special theory of relativity......Page 340
12. Relativistic dynamics......Page 345
13. The general theory of relativity......Page 348
1. Definitions......Page 357
2. Vector differential forms and surface theory......Page 364
3. Scalar and vector products of vector forms on surfaces and their exterior derivatives......Page 371
4. Some formulas for closed surfaces. Characterizations of the sphere......Page 373
5. Minimal surfaces......Page 378
6. Uniqueness theorems for closed convex surfaces......Page 380
2. Geometry in an affine space......Page 393
3. Tensor algebra in centered affine spaces......Page 397
4. Effect of a change of basis......Page 400
5. Definition of tensors......Page 402
6. Tensor algebra in Euclidean spaces......Page 407
1. Theorems on ordinary differential equations......Page 410
2. Overdetermined systems of linear partial differential equations......Page 414
Bibliography......Page 418
Index......Page 423