|
Abstract |
9-11 |
|
1.Introduction |
11-17 |
|
1.1 Some issues in geometry |
11-13 |
|
1.1.1 Differential geometry |
11-12 |
|
1.1.2 Curvature |
12-13 |
|
1.1.3 Convexity |
13 |
|
1.2 Motivation |
13-15 |
|
1.2.1 On the theorem of Alexandrov |
13-14 |
|
1.2.2 On invariants of the intersection of surfaces |
14-15 |
|
1.3 Organization of the thesis |
15-17 |
|
2.Euclidean space |
17-23 |
|
2.1 Topologies and open/closed sets |
17-19 |
|
2.2 Connectedness and Compactness |
19 |
|
2.3 Immersion and Imbedded |
19-21 |
|
2.4 Linear group |
21-23 |
|
3.Curves and surfaces |
23-32 |
|
3.1 Basic concepts of curves |
23-25 |
|
3.2 Tangent vectors |
25-26 |
|
3.3 Gauss curvature and mean curvature |
26-27 |
|
3.4 Geodesics |
27-29 |
|
3.5 Global differential geometry |
29-32 |
|
4.Proofs of the theorems |
32-40 |
|
4.1 On the Alexandrov's theorem |
32-36 |
|
4.1.1 Several concepts and lemmas |
32-34 |
|
4.1.2 Proof of the theorem |
34-36 |
|
4.2 On invariants of the intersection of surfaces |
36-40 |
|
4.2.1 Useful lemma |
36-37 |
|
4.2.2 Proof of the theorem |
37-40 |
|
5.Conclusion |
40-42 |
|
5.1 On the Alexandrov's theorem |
40-41 |
|
5.2 On the formula of curvature |
41-42 |
|
Acknowledgements |
42-43 |
|
Bibliography |
43-46 |
