The lie brocket

Can you think of a property that a connection has, that the Lie bracket for vector fields doesn't? The definition of "connection" doesn't list that many properties, so you can check them one at a time. For example, a connection is ℝ-linear in both variables, and so is the Lie bracket for vector fields. That doesn't help, so you need to keep checking until you find a property that this Lie bracket doesn't have.

 

Yes, but I don't want to do your work for you. If you post the properties that you know a connection must have, and your attempts to determine if a Lie bracket has those properties too, I will tell you if you're doing something wrong. If you get stuck, show me where, and I'll try to help you get past that point.

 

Can you evaluate the following expressions, where X and Y are arbitrary vector fields, and f is an arbitrary smooth function?

[tex]\nabla_{X}(fY)(p)[/tex]

[tex][X,fY]_p[/tex]

Click the quote button, and you'll see how I did the LaTeX. If you try it, you need to keep in mind that there's a bug that makes the wrong images appear. The only workaround is to refresh and resend after each preview.

The policy around here is that people who ask for help with textbook-style problems have to show a complete statement of the problem, the definitions they're using, and their work so far, so that we can give hints that will help them move past the point where they are stuck.

I don't understand what you're asking. what is V(t)? What do you mean by 6(u,v)? What manifold are you talking about, and what metric/connection are you using?

 

In LaTeX, you need to use \ instead of /. For example, \sigma instead of /sigma. I also recommend \cos instead of cos.

I don't have time to look at your problem today. Maybe someone else does.