# 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.