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 in the following context:

Consider two manifolds and and a mapping of the prior into the latter. Then for a point , the

differentialof at is a linear mapping which is defined as follows: Given a vector , choose a path with . Then is the vector tangent to the curve at .

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 be a manifold of dimension and let be a -form on for . Here, 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 to the -form where , where is the multi-index consisting of all numbers not in , and where denotes the sign of .

Here’s what all that says:

Say we’re in with the standard orthonormal directional 1-forms , , and . As expected, the 0-form gets mapped to and the volume form satisfies . Also, the Hodge star operator maps each of the aforementioned coordinate 1-forms to the following 3-1 = 2-forms:

, , and .

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

, , .

As a less trivial example^{1}, consider the case of (again) with the metric . This space + metric combination is the Lorentzian manifold known as *Minkowski Space*. Here, the basis has the form but the presence of the negative square of the component changes the behavior of the Hodge star operator as seen below:

.

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.

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