For this surface M, I guess I need to show M has no boundary, and find the euler characterstic of M. How to do this?

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- May 7th 2013, 04:58 AMPlato13Gauss bonnet theorem
For this surface M, I guess I need to show M has no boundary, and find the euler characterstic of M. How to do this?

- May 7th 2013, 06:48 AMxxp9Re: Gauss bonnet theorem
You can show that M is homeomorphic to S^2 so it has no boundary and its Euler character is 2.

To show that M is homeomorphic to S^2={(x,y,z)|x^2+y^2+z^2=1}, define a map: f: M -> S^2 by f(p) = p/|p| and try to show that f is a homeomorphism( actually it is a diffeomorphism). - May 7th 2013, 07:29 AMPlato13Re: Gauss bonnet theorem
Ok thanks, so the total curvature is 2 x 2pi=4pi. What about (ii)

- May 7th 2013, 07:43 AMxxp9Re: Gauss bonnet theorem
(ii) is a direct application of the theorem. -4 pi = 2 pi X, so X=-2. It is a double torus.