Hodge Dual part deux

You may recall that I deemed yesterday Differential Geometry Sunday and posted a small expository thing on the Hodge Star/Dual operator. Apparently in my cloudy haze of mathematical mediocrity, I concluded my post without having touched on the derivations I actually intended to touch on.

Sometimes I feel like I need a vacation from my vacation.

In any case, I’m going to take a stab at saying some of the things I’d meant to say yesterday, but in order to ensure we’re all on the same page, I’m going to recall what exactly the Hodge Star/Dual operator is. Then, after the break, I’m going to show some of the cool derivations that come about because of it.

Let M be a manifold of dimension \dim(M)=n and let \omega=\sum_I f_I dx_I be a k-form on M for k\leq n. Here, I denotes a multi-index which – without loss of generality – can be assumed to be increasing. Then the hodge star operator is the operator which takes \omega to the (n-k)-form *\omega =\sum_I f_I(*dx_I) where *dx_I=\varepsilon_I dx_{I^C}, where I^C is the multi-index consisting of all numbers 1,\ldots,n not in I, and where \varepsilon_I=\pm 1 denotes the sign of dx_I dx_{I^C}.

So what we said is that for the most commonly-recognized example \mathbb{R}^3 with orthogonal 1-forms dx,dy,dz and the usual metric ds^2=dx^2+dy^2+dz^2, the Hodge Star operator sends 1 to dx\wedge dy\wedge dz and vice versa, it sends dx,dy,dz to dy\wedge dz, dz\wedge dx,dx\wedge dy respectively, and it sends (dx\wedge dy)\mapsto dz, (dy\wedge dz)\mapsto dx, and (dx\wedge dz)\mapsto -dy. But the question then remains: Why does anybody care?

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Differential Geometry Sunday

I was able to actually stick with my plan a little earlier and spend the day parsing through some stuff in Kobayashi and Nomizu. That’s not a bad way to spend a Sunday.

Somewhere in the middle of that, I ended up stumbling upon something I’d always been somewhat privy to and I did so almost by accident. In the text, I ran across the definition f_* in the following context:

Consider two manifolds M and M' and a mapping f:M\to M' of the prior into the latter. Then for a point p\in M, the differential of f at p is a linear mapping f_*:T_p(M)\to T_{f(p)}(M') which is defined as follows: Given a vector X\in T_p(M), choose a path x(t) with p=x(t_0). Then f_*(X) is the vector tangent to the curve f(x(t)) at f(p)=f(x(t_0)).

The notation reminded me of something I saw during my very first foray into Differential Geometry, namely the Hodge star/dual operator. It was a notion that was so novel when I first saw it that I contemplated preparing a seminar talk at BGSU for my peers, none of whom were geometers of any kind; now that I’ve rediscovered it, I’m having similar ideas for my non-geometer peers here at FSU. But I’m getting ahead of myself…

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