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?
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).