One week in

So, I’ve officially survived the first week of my (second) second year as a Ph.D. student. It’s hard to imagine sometimes that I’ve been a grad student for over three years.

I’m such an old man.

It’s been even less easy than I’d anticipated, too, which is nicht spaß; sadly, too, this is the first week where teaching responsibilities are actually a thing (we get week 1 off from such things), so now it’s difficult and time-consuming. Jajajajaja.

Rather than making this entry all about Yours Truly, I figured I’d pit stop in and use some of my (*gasp*) unoccupied time write up a little spiel I saw for the first time in my Riemannian geometry class last week.

In topology, there’s an idea called invariance of dimension which can be stated in many different contexts, situations, etc. This can be modified in the case of manifolds with differential structures, and because the idea of the proof seemed a bit cool, I decided to throw it up here for you guys. Throughout, the notation M^m denotes a manifold M of dimension m with an associated differential structure.

Claim. If M^m and N^n are diffeomorphic, then m=n.
Proof. Let (U,X), respectively (V,Y) be a differential structure on M^m = M, respectively N^n=N and let \varphi:M\to N be a diffeomorphism. Because compositions of differentiable mappings are all differentiable, it follows that both

Y^{-1}\circ\varphi\circ X:U\subset\mathbb{R}^m\to V\subset\mathbb{R}^n and X^{-1}\circ\varphi^{-1}\circ Y:V\to U

are differentiable mappings.

From a well-known fact concerning differentiable manifolds, the differential map \varphi:M\to N induces at each point p\in M a linear map d\varphi_p (called the differential of \varphi at p) between the associated tangent spaces T_pM and T_{\varphi(p)}N which is actually linear. It follows, then, that the composition Y^{-1}\varphi X induces a linear map \Psi of the form

\Psi:T_{X^{-1}(p)}\mathbb{R}^m\to T_{Y\varphi(p)}\mathbb{R}^n;

similarly, X^{-1}\varphi^{-1}Y induces a linear map \Phi,

\Phi:T_{Y\varphi(p)}\mathbb{R}^n\to T_{X^{-1}(p)}\mathbb{R}^..

Moreover, one can easily verify that \Psi, \Phi are inverses of one another, whereby it follows that the aforementioned tangent spaces are isomorphic. Therefore, it follows that

\mathbb{R}^m\cong T_{X^{-1}(p)}\mathbb{R}^m\cong T_{Y\varphi(p)}\mathbb{R}^n\cong\mathbb{R}^n,

and as a result of the usual invariance of domain from point-set topology (provable by Brouwer’s fixed point theorem, among others), \mathbb{R}^m\cong\mathbb{R}^n if and only if m=n. Hence, the result. \square

This was a bit of a cheapo entry: Not too involved, not too difficult. Maybe next time, I’ll have something more meaningful to share…or maybe I’ll just come here and gripe about how difficult it is to set up a Scientific Linux 6 workstation the way I want it from scratch.

Or maybe not.

Au revoir!

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