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 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 .
So what we said is that for the most commonly-recognized example with orthogonal 1-forms and the usual metric , the Hodge Star operator sends to and vice versa, it sends to respectively, and it sends , , and . But the question then remains: Why does anybody care?
From a big picture perspective, mathematicians are all about generality and the reason is obvious: Why should I have to resolve problems time and again using specific information when generalities will allow me to solve multiple cases at one time without having to do repeat a process indefinitely? It makes sense, as much as our brains hate to admit it, and mathematicians are all about making sense.
Differential geometry is, in many ways, an offspring of this mentality. One way to think about the advancement towards modern differential geometry is as follows: Learn one-variable calculus, use basic intuition to advance to multi-variable vector calculus, and then ask yourself, What if I want to do calculus outside the confines of simple Euclidean spaces? What if I want to do calculus on all smooth surfaces at once? The solution, of course, is differential geometry, with its use of differential forms to generalize standard Euclidean infinitesimals and its three main operations (namely , , and ) to mimic the behavior of differentiation and integration in far more general environments.
So, long story short: The Hodge Dual should be thought of as one of three main operations, and so its properties are studied in part because they’re intrinsically beautiful and in part because the interplay between it and its two cousin operations is really really nice.
And that’s the purpose of this post.
Consider the 1-form where for the time being we’re working in . Clearly, then, can be thought of as a vector-valued 1-form of the form where and . This yields an obvious one-to-one correspondence between 1-forms and vector fields , and this correspondence yields some particularly interesting results when hit with the Hodge Star. To see this, let be an arbitrary smooth function (i.e., a 0-form) with associated 1-form . Rewriting, we see that
The kicker, here, is that the vector field is precisely the gradient field . Hence,
Next, consider a vector field and a 1-form . As mentioned above,
where means “omitting ” and where the second equation comes from rewriting the -form in increasing multi-index notation. This notation can be simplified by writing for the vector-valued -form of the form
and by noting then that . Taking the exterior derivative of yields the following:
Recollecting the vector field as the so-called divergence field , whereby it follows that
Note, also, that this identity can be rewritten by observing that in , whereby it follows that . Hence,
Finally, note than in , evaluating and then applying yields that
The details of the last identity are omitted for brevity but can be found – along with further details of the identities previously mentioned – in .
So there we have it: A (more-complete) (somewhat-less-)brief exposition of reasons why differential geometers love the Hodge Star operator. Of course, this last statement is an absurdly-over-simplified over-simplification, but for now, it summarizes where we stand with our knowledge of the topic at hand.
Until next time….
1. Sjamaar, Reyer, Manifolds and Differential Forms. Retrieved online from here.