Another proof that a vector field on the sphere must have a zero?

[lavinia]

I.e. you still need to prove such an extension exists, plus you must prove that all vector fields have the same total degree. It seems to me this is far more than needed for the result, as Lefschetz's simple argument shows. I like seeing that they are in essence the same though.

I was not thinking about the extension. The point was that you easily see the attaching map of the two solid tori from this vector field.

 

[lavinia]

by the way thanks very much for the discussion. i now understand how lefschetz's hands on proof fits into these other ideas MUCH more than I did before.

I think this thread was highly instructive. I enjoy talking with you. I always learn something from you.

 

[mathwonk]

thank you lavinia. I too get a lot of enjoyment from your posts. I think you are quite strong and I am sure you have a bright future in any area you choose. Do you get all this from reading? Or have you also taken some good classes?

I suggest you look at mathoverflow if you have not. It is more of a grad student/professional level math site and I have learned a lot just from reading the questions and answers there.

 

[mathwonk]

apparently i have not read your discussion of solid tori. my apologies. i am somewhat slow to learn. i like to think of my own solutions. i don't know a lot, but i understand fairly well the things i do know.

 

[lavinia]

thank you lavinia. I too get a lot of enjoyment from your posts. I think you are quite strong and I am sure you have a bright future in any area you choose. Do you get all this from reading? Or have you also taken some good classes?

I suggest you look at mathoverflow if you have not. It is more of a grad student/professional level math site and I have learned a lot just from reading the questions and answers there.

I am in awe that you knew Lefschetz. He is one of the immortals.

I mostly read - lately papers - but have sat in on a few classes. what do you do mathematically?

 

[lavinia]

Related to all of this is the differential geometry of the sphere. When it has constant curvature,1, the exterior derivative of its connection 1 form on the unit circle bundle is the pullback of its volume form under the bundle projection map

If there were a non zero vector field, we could normalize it to have length 1 to get a non-zero section of the unit circle bundle. Then the volume form of the sphere would have to be exact. This is the same contradiction one gets from knowing that the tangent circle bundle is RP^3.

The same arguments apply to other surfaces except for the torus since they all can be given geometries with constant negative curvature.

It also follows that any metric on the torus must have points of zero curvature.

So the connection 1 form again tells you that the tangent circle bundle is not a trivial bundle. How does this link up with everything else we have been talking about?

Also the connection 1 form reminds me of the differential form approach to linking number.

 

[mathwonk]

I only read his work, and saw a movie featuring him. That movie is still available I would guess, from the MAA?

well i cannot locate a copy of that movie but the library at cornell has the ones of bott and marston morse, but those as i recall are not as entertaining.