Geometrical methods of mathematical physics

[pmb_phy]

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

...there is no C^inf 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, ...

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 C^inf vector field on S² which is nowhere zero.

Help?

Pete

 

[HallsofIvy]

That's known as the "hairy ball" theorem. TIf you imagine a ball (or someone's head) completely covered with hair, there must be a "whorl" or a place where there is no hair. Basically, if you start with a continuous, non-zero, vector field at a point and follow along both directions of great circles around the ball, at some point, the directions will conflict.

If you are asking "how they reach the conclusion that there is no C^inf vector field on S² which is nowhere zero" in that quote, they don't, they simply assert it. If memory serves, that is a difficult theorem to prove.

 

[selfAdjoint]

Likewise consider the atmosphere of the earth; the theorem states that at any given moment, somewhere on earth, the wind is calm.

The reason is basically that if you had a vector field with no zero, you could use it to construct a shrinking of the sphere, in itself, to a point, and this is impossible; the 2-sphere is o-connected (i.e you can get from any point on it to any other by a continuous path). And it is 1-connected (i.e you can shrink any closed curve, in the sphere, to a point), but it is not 2-connected.

 

[Hurkyl]

A nowhere-zero vector field would generate such a map with no fixed point, ...

It's saying that you find some way to use your vector field to define a family of continuous maps from the sphere to itself. You could imagine the vector field as describing something sophisticated like a flow... or something simpler like "this point moves that-a-way".

Because the sphere is compact, that guarantees you can find a parameter sufficiently small so that the only way a point could be mapped to itself is if the corresponding tangent vector at that point is zero.