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pmb_phy
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I'm reading Schutz's text Geometrical methods of mathematical physics right now as a part of a diredcted study (Two of my former math professors and I wish to learn the subject together so we formed a group for self-study). I know the material in a general sense but not at the precision of this wonderful text. But there is a comment that I don't understand on page 39 which, regarding the tangent-bundle on a 2-sphere, states that
Help?
Pete
Okay. I may be burned out on this stuff by now but frankly I don't see how they reach the conclusion that there is no Cinf vector field on S2 which is nowhere zero....there is no Cinf vector field on S2 which is nowhere zero. This is a consequence of the famous but difficult fixed-point theorem of the sphere, that every 1-1 map (difffeomorphism) of S2 onto itself leaves one point fixed, provided that the map is a member of a continuous family of maps containing the identity map. A nowhere-zero vector field would generate such a map with no fixed point, ...
Help?
Pete