Today is the first day of the third week of the semester. I know that counting down like this is going to make it seem longer than it already seems, but it seems so long that I can’t seem to help remaining conscious of the precise time frame I’m dealing with.
Such is life, I suppose.
I’ve noticed an amusing trend in my page views involving solutions from Dr. Hatcher’s book, namely that I’ve been receiving an abnormally-high level of page views lately, almost all of which seem to center on those solutions. I guess that means that the semester has started elsewhere too and that people find topology as difficult and frustrating as I do.
For those of you who fit this bill and who are reading this right now: My plan is to start doing more problems ASAP, so that page might get its first update in quite a while.
Today, though, I want (read: need) to talk about differential geometry. In particular, we spent some time in class last week discussing the Lie bracket and its properties, and because we have a derivation of one particular property, I wanted to take the time to put that here for my own benefit.
Given a differentiable manifold of dimension with coordinate chart , two differentiable vector fields and on can be written in terms of the related local coordinates as follows:
where are smooth functions. For brevity, let denote . Note, then, that one can always write down a formula for, say, :
On the other hand, the resulting operator may not be a vector field due (roughly) to the existence of the second-order derivatives in the resulting expression. Therefore, the Lie bracket is a mechanism for remedying that.
Definition. Given two differentiable vector fields and on as above, the Lie bracket is an operator which assigns to this pair the operator which is guaranteed to be a vector field on .
The Lie bracket has a whole slew of nice, useful, or otherwise worthwhile properties. The one I’m here to talk about today is as follows: Given differentiable functions ,
Now one of the things I’ve always struggled with in this field is notation, so here’s a brief rundown: Because is a function and is a vector field (both on ), the quantity acts on an arbitrary point as . This is a vector. Similarly, is a function which assigns to each point the point . So then, we can now begin:
Proof. By definition of the Lie bracket,
Therefore, it suffices to determine what each of these summands does, and perhaps the easiest way is to determine directly how the operator acts on functions. To that end, let be a differentiable function. One can easily show (based on the properties of derivations, say) that for arbitrary functions , (i.e., the Leibniz product rule). Therefore, it follows that . Substituting into , it follows that
Because this identity holds for arbitrary, the identity holds for all differentiable functions from to and hence the result is proved. Note that the step in stems from the Leibniz rule as noted above.
I can say without a doubt that this topic is absurdly notation-heavy, but I haven’t not-loved a single day of the class so far.
Excitement: I haz it.