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…

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}.

Here’s what all that says:

Say we’re in \mathbb{R}^3 with the standard orthonormal directional 1-forms dx, dy, and dz. As expected, the 0-form 1 gets mapped to *1=dx\wedge dy\wedge dz and the volume form dx\wedge dy\wedge dz satisfies *(dx\wedge dy\wedge dz)=1. Also, the Hodge star operator * maps each of the aforementioned coordinate 1-forms to the following 3-1 = 2-forms:

*dx=dy\wedge dz, *dy=dz\wedge dx, and *dz=dx\wedge dy.

Similarly, the following summarizes the behavior of the Hodge star on the 2-forms:

*(dx\wedge dy)=dz, *(dx\wedge dz)=-dy, *(dy\wedge dz)=dx.

As a less trivial example1, consider the case of \mathbb{R}^3 (again) with the metric ds^2=dx^2+dy^2-dt^2. This space + metric combination is the Lorentzian manifold known as Minkowski Space. Here, the basis has the form \{dx,dy,dt\} but the presence of the negative square of the t component changes the behavior of the Hodge star operator as seen below:

*1=dx\wedge dy\wedge dt
*dx = dy\wedge dt
*dy = -(dx\wedge dt)=dt\wedge dx
*dt = dx\wedge dy
*(dx\wedge dy) = dt
*(dy\wedge dt)=-dx
*(dx\wedge dt)=-dy
*(dx\wedge dy\wedge dt)=-1.

As expected, the properties of the Hodge star operator run far more deeply than the ones described above. To see more, see the paper cited in [1] below. There are also lots of other resources online as well.

    1. Example taken from Tevian Dray of Oregon State.

One thought on “Differential Geometry Sunday

  1. Pingback: Hodge Dual part deux | riemannian hunger

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