《An Introduction To Differential Geometry》求取 ⇩

PART 1THE THEORY OF CURVES AND SURFACES IN THREE-DIMENSIONAL EUCLIDEAN SPACE1

Ⅰ．THE THEORY OF SPACE CURVES1

1．Introductory remarks about s pace curves1

2．Definitions3

3．Arc length5

4．Tangent，normal，and binormal7

5．Curvature and torsion of a curve given as the intersection of two surfaces16

6．Contact between curves and surfaces18

7．Tangent surface，involutes，and evolutes21

8．Intrinsic equations，fundamental existence theorem for space curves23

9．Helices26

Appendix Ⅰ．1．Existence theorem on linear differential equations27

Miscellaneous Exercises Ⅰ29

Ⅱ．THE METRIC:LOCAL INTRINSIC PROPERTIES OF A SURFACE31

1．Definition of a surface31

2．Curves on a surface35

3．Surfaces of revolution36

4．Helicoids37

5．Metric39

6．Direction coefficients41

7．Families of curves44

8．Isometric correspondence48

9．Intrinsic properties52

10．Geodesics54

11．Canonical geodesic equations59

12．Normal property of geodesics62

13．Existence theorems65

14．Geodesic parallels69

15．Geodesic curvature70

16．Gauss－Bonnet theorem75

17．Gaussian curvature78

18．Surfaces of constant curvature81

19．Conformal mapping83

20．Geodesic mapping87

Appendix Ⅱ．1．The second existence theorem90

Miscellaneous Exercises Ⅱ92

Ⅲ．THE SECOND FUNDAMENTAL FORM:LOCAL NON-INTRINSIC PROPERTIES OF A SURFACE95

1．The second fundamental form95

2．principal curvatures97

3．Lines of curvature99

4．Developables101

5．Developables associated with space curves103

6．Developables associated with curves on surfaces105

7．Minimal surfaces106

8．Ruled surfaces107

9．The fundamental equations of surface theory111

10．Parallel surfaces116

11．Fundamental existence theorem for surfaces119

Miscellaneous Exercises Ⅲ124

Ⅳ．DIFFERENTIAL GEOMETRY OF SURFACES IN THE LARGE127

1．Introduction127

2．Compact surfaces whose points are umbilics128

3．Hilbert＇s lemma129

4．Compact surfaces of constant Gaussian or mean curvature131

5．Complete surfaces131

6．Characterization of complete surfaces133

7．Hilbert＇s theorem137

8．Conjugate points on geodesics145

9．Intrinsically defined surfaces151

10．Triangulation154

11．Two-dimensional Riemannian manifolds157

12．The problem of metrization159

13．The problem of continuation162

14．Problems of embedding and rigidity164

15．Conclusion164

PART 2DIFFERENTIAL GEOMETRY OF n-DIMENSIONAL SPACE166

Ⅴ．TENSOR ALGEBRA166

1．Vector spaces166

2．The dual space169

3．Tensor product of vector spaces172

4．Transformation formulae179

5．Contraction183

6．special tensors185

7．Inner product188

8．Associated tensors188

9．Exterior algebra189

Miscellaneous Exercises Ⅴ192

Ⅵ．TENSOR CALCULUS193

1．Differentiable manifolds193

2．Tangent vectors195

3．Affine tensors and tensorial forms200

4．Connexions205

5．Covariant differentiation209

6．Connexions over submanifolds216

7．Absolute derivation of tensorial forms．218

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

Appendix Ⅵ．2．Tensor-connexions223

miscellaneous Exercises Ⅵ224

Ⅶ．RIEMANNIAN GEOMETRY226

1．Riemannian manifolds226

2．Metric226

3．The fundamental theorem of local Riemannian geometry228

4．Differential parameters231

5．Curvature tensors232

6．Geodesics233

7．Geodesic curvature235

8．Geometrical interpretation of the curvature tensor236

9．Special Riemannian spaces237

10．Parallel vectors239

11．Vector subspaces240

12．Parallel fields of planes242

13．Recurrent tensors245

14．Integrable distributions249

15．Riemann extensions258

16．E．Cartan＇s approach to Riemannian geometry261

17．Euclidean tangent metrics266

18．Euclidean osculating metrics268

19．The equations of structure271

20．Global Riemannian geometry273

Bibliographies on harmonic spaces，recurrent spaces，parallel distributions，Riemann extensions276

miscellaneous Exercises Ⅶ278

Ⅷ．APPLICATIONS OF TENSOR METHODS TO SURFACE THEORY281

1．The Serret－Frenet formulae281

2．The induced metric284

3．The fundamental formulae of surface theory287

4．Normal curvature and geodesic torsion290

5．The method of moving frames294

Miscellaneous Exercises Ⅷ299

EXERCISES300

SUGGESTIONS FOR FURTHER READING314

INDEX315