An Introduction To Differential GeometryPDF电子书下载

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PART 1 THE THEORY OF CURVES AND SURFACES IN THREE-DIMENSIONAL EUCLIDEAN SPACE 1

Ⅰ．THE THEORY OF SPACE CURVES 1

1．Introductory remarks about s pace curves 1

2．Definitions 3

3．Arc length 5

4．Tangent，normal，and binormal 7

5．Curvature and torsion of a curve given as the intersection of two surfaces 16

6．Contact between curves and surfaces 18

7．Tangent surface，involutes，and evolutes 21

8．Intrinsic equations，fundamental existence theorem for space curves 23

9．Helices 26

Appendix Ⅰ．1．Existence theorem on linear differential equations 27

Miscellaneous Exercises Ⅰ 29

Ⅱ．THE METRIC:LOCAL INTRINSIC PROPERTIES OF A SURFACE 31

1．Definition of a surface 31

2．Curves on a surface 35

3．Surfaces of revolution 36

4．Helicoids 37

5．Metric 39

6．Direction coefficients 41

7．Families of curves 44

8．Isometric correspondence 48

9．Intrinsic properties 52

10．Geodesics 54

11．Canonical geodesic equations 59

12．Normal property of geodesics 62

13．Existence theorems 65

14．Geodesic parallels 69

15．Geodesic curvature 70

16．Gauss－Bonnet theorem 75

17．Gaussian curvature 78

18．Surfaces of constant curvature 81

19．Conformal mapping 83

20．Geodesic mapping 87

Appendix Ⅱ．1．The second existence theorem 90

Miscellaneous Exercises Ⅱ 92

Ⅲ．THE SECOND FUNDAMENTAL FORM:LOCAL NON-INTRINSIC PROPERTIES OF A SURFACE 95

1．The second fundamental form 95

2．principal curvatures 97

3．Lines of curvature 99

4．Developables 101

5．Developables associated with space curves 103

6．Developables associated with curves on surfaces 105

7．Minimal surfaces 106

8．Ruled surfaces 107

9．The fundamental equations of surface theory 111

10．Parallel surfaces 116

11．Fundamental existence theorem for surfaces 119

Miscellaneous Exercises Ⅲ 124

Ⅳ．DIFFERENTIAL GEOMETRY OF SURFACES IN THE LARGE 127

1．Introduction 127

2．Compact surfaces whose points are umbilics 128

3．Hilbert＇s lemma 129

4．Compact surfaces of constant Gaussian or mean curvature 131

5．Complete surfaces 131

6．Characterization of complete surfaces 133

7．Hilbert＇s theorem 137

8．Conjugate points on geodesics 145

9．Intrinsically defined surfaces 151

10．Triangulation 154

11．Two-dimensional Riemannian manifolds 157

12．The problem of metrization 159

13．The problem of continuation 162

14．Problems of embedding and rigidity 164

15．Conclusion 164

PART 2 DIFFERENTIAL GEOMETRY OF n-DIMENSIONAL SPACE 166

Ⅴ．TENSOR ALGEBRA 166

1．Vector spaces 166

2．The dual space 169

3．Tensor product of vector spaces 172

4．Transformation formulae 179

5．Contraction 183

6．special tensors 185

7．Inner product 188

8．Associated tensors 188

9．Exterior algebra 189

Miscellaneous Exercises Ⅴ 192

Ⅵ．TENSOR CALCULUS 193

1．Differentiable manifolds 193

2．Tangent vectors 195

3．Affine tensors and tensorial forms 200

4．Connexions 205

5．Covariant differentiation 209

6．Connexions over submanifolds 216

7．Absolute derivation of tensorial forms． 218

Appendix Ⅵ．1．Tangent vectors to manifolds of class ？ 221

Appendix Ⅵ．2．Tensor-connexions 223

miscellaneous Exercises Ⅵ 224

Ⅶ．RIEMANNIAN GEOMETRY 226

1．Riemannian manifolds 226

2．Metric 226

3．The fundamental theorem of local Riemannian geometry 228

4．Differential parameters 231

5．Curvature tensors 232

6．Geodesics 233

7．Geodesic curvature 235

8．Geometrical interpretation of the curvature tensor 236

9．Special Riemannian spaces 237

10．Parallel vectors 239

11．Vector subspaces 240

12．Parallel fields of planes 242

13．Recurrent tensors 245

14．Integrable distributions 249

15．Riemann extensions 258

16．E．Cartan＇s approach to Riemannian geometry 261

17．Euclidean tangent metrics 266

18．Euclidean osculating metrics 268

19．The equations of structure 271

20．Global Riemannian geometry 273

Bibliographies on harmonic spaces，recurrent spaces，parallel distributions，Riemann extensions 276

miscellaneous Exercises Ⅶ 278

Ⅷ．APPLICATIONS OF TENSOR METHODS TO SURFACE THEORY 281

1．The Serret－Frenet formulae 281

2．The induced metric 284

3．The fundamental formulae of surface theory 287

4．Normal curvature and geodesic torsion 290

5．The method of moving frames 294

Miscellaneous Exercises Ⅷ 299

EXERCISES 300

SUGGESTIONS FOR FURTHER READING 314

INDEX 315

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