- #1
joseph0086
- 4
- 0
Hello, all, the most important results that I know in this topic is the Gauss-Bonnet Theorem (and hence the classification of compact orientable surfaces) and also the Poincare-Hopf index theorem.
But there are still some fundamental problems I don't understand.
For example, is the following problem true?
Let C be a simple closed curve on a sphere. Let v be a vector field on it such that v is never tangent to C. Is it true that each of the regions determined by C contains at least one singular point of v.
I feel it is true, but I don't know how to prove it...
Moreover, it is intuitivlely clear that all compact orientable surfaces with negative euler characteristics have some regions which have Gaussian curvatures positive, negative and zero. The fact that it has regions that have -ve curvature is obvious by Guass-Bonnet. But how to show mathematically that it must have some regions that have positive curvature? (In other words, my question is: Show that compact orientable surface has negative curvature at all points.)
I learned from a textbook that for all such surfaces with -ve Euler characteristics, then two geodesics which start from the same point won't meet again. (Easy to show by Guass-Bonnet), I don't know if it is related to my question..
Hope somebody here can tell me his understanding.. Thanks a lot.
But there are still some fundamental problems I don't understand.
For example, is the following problem true?
Let C be a simple closed curve on a sphere. Let v be a vector field on it such that v is never tangent to C. Is it true that each of the regions determined by C contains at least one singular point of v.
I feel it is true, but I don't know how to prove it...
Moreover, it is intuitivlely clear that all compact orientable surfaces with negative euler characteristics have some regions which have Gaussian curvatures positive, negative and zero. The fact that it has regions that have -ve curvature is obvious by Guass-Bonnet. But how to show mathematically that it must have some regions that have positive curvature? (In other words, my question is: Show that compact orientable surface has negative curvature at all points.)
I learned from a textbook that for all such surfaces with -ve Euler characteristics, then two geodesics which start from the same point won't meet again. (Easy to show by Guass-Bonnet), I don't know if it is related to my question..
Hope somebody here can tell me his understanding.. Thanks a lot.