...Prove Their Worth...

"Problems worthy of attack
prove their worth
by hitting back." - Piet Hein

A kind of running diary and rambling pieces on my struggles with assorted books, classes, and other things, as they happen. You must be pretty bored to be reading this...

Thursday, June 20, 2002

The promised details on Theorem 9, Ch2:



First, some terminology. In differential geometry the objects studied are manifolds. A manifold is just a space that looks like R^n if you look at any little part of it. R^n is just normal flat space, as R is just the real line, and the n stands for its dimension (n = 3 for the space we live in*). In general, a manifold can look like R^n's of differing n's in different areas, but we can ignore that for the most part.



There's lots of theorems one can play with that use only that much information, but to get to the 'differential' part of differential geometry, we need some more structure. For instance, we might want to assign some coordinates to our manifold, M, or at least to a small part of it. To do that that, we look at some subset U of M. Then we define some function, x, which, given a point in U, will spit out a point in R^n. Just to be fancy, we call (x, U) an atlas of M. We can show that this definition makes sense, and meshes with our normal conception of coordinates as just a grid.



Armed with this idea, and a few others I'm not going to go into here, we can start doing calculus on manifolds. We can define functions between manifolds, and take their derivatives. This is actually a very neat process.



Let's call our function f, and have it take points in M^n to points in another manifold, A^m. To actually do this, we involve some unspecified charts on both manifolds, say (x, U) on M, and (y, V) on A. Now, remember, x takes M -> R^n. So x^-1 takes R^n -> M. (Same goes for y, of course.) The thing is that it's really pretty simple to define functions that do things to R^n - we've all been doing it since middle school (except we didn't call it R^n back then ...) So, let's stick with what we know, and try to make f work in R^n (or between R^n and R^m - whatever), but also in the process do what we need it to do, which is work with manifolds.



Here's how: It's just y*f*x^-1, with * meaning composition of functions, y, f, and x^-1 being functions (duh), and the whole thing is to be read right to left, as one normally (!) does with composition of functions. So, given a point q in M, we want to get a point f(q) in A. Ok. Let's feed q to x^-1, getting x^-1[q] (notice, now we're in R^n!). Now, feed that to f, getting f[x^-1[q]] (notice, we fed f R^n, and it shat out R^m - so it's a good ol' function, the kind we know how to handle). And now we feed that to y, getting y[f[x^-1[q]]]. Notice that y is what takes us from R^m to where we wanted to get, which is the manifold A^m !



Whew.



So, look at what we did: we 'hid' the fact that we're actually feeding it with something bizarre, like manifolds, from f, and persuaded it that it's actually muching on simple tasty things like R^n. So while we're lying unscrupulous bastards, we got what we wanted: f really is a function between manifolds. It just so happens it needs to wear blinders to do it, otherwise it would run away in terror.



If you followed that, I hate you and envy you, because it took me several days to get that far.



Now, since we can define these functions, and they're continuous, blah blah blah, we can take their derivatives, and we can talk about 'changing coordinates', and crap like that. We can even define big-ass Jacobians, which are matrices that tend to pop up when you squint carefully at the idea of changing from one set of coordinates to another. And we can play with matrix, figuring out its 'rank' and other linear-alebraish things. (Note: it's rank is going to vary from place to place!)



So, err, that was the introduction. The theorem (Th. 9, Ch2, Spivak's DG) says: Say we've got a function f that takes things from one manifold, M^m, to another, A^n. Say further that it has a rank k at the point p in M^n. Then, given p = (a_1, a_2, ..., a_m):



y*f*x^-1[a_1, a_2, ..., a_m] = (a_1, a_2, ..., a_k, phi_(k+1), phi_(k+2), ..., phi_(n))


(the phi's are just numbers you get from f in a certain way, as it turns out, and I've left out the second part of the theorem, which looks the same except for having a bunch of zeros instead of the phi's.).



Now, what that is saying are some things about what actually happens if you stuff one manifold into another: depending on f and p, some of parts of your points are going to remain fixed and other parts are going to get chewed on. Different points are going to contain different amounts of dietary undigested fiber**, if you will. Hopefully, that makes some amount of sense.



I didn't understand this at all until late last night, when, after having a few drinks at a birthday party (not mine), I was sitting upon the Throne of Power. There, pretty much all of the above hit me. I was stuck on this damn theorem for three days straight, though as I understand now, that was because I didn't correctly grasp some theorems before.



Now, I intentionally glossed over quite a few things in the above exposition, there are parts (close to all of them) where I have a suspicion I don't really know what I'm talking about, and there are parts where I'm probably saying something terribly naive and stupid or both. That is the curse and the blessing of studying alone: there's no one to yell "Hey, you bloody stupid arse-scratcher, that's wrong and dumb!"



* - Well, 4, if you want to be a laxative butt-monkey about it.


** - I don't think manifolds are part of the food pyramid, but they should be!

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