Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Hodge operator and adjoints

  1. Feb 28, 2008 #1
    Usually the adjoint to the exterior derivative [tex]d^*[/tex] on a Riemannian manifold is derived using the inner product
    where [tex]\lambda[/tex] are p-forms and [tex]*[/tex] is the Hodge duality operator taking p-forms to (n-p)-forms which is defined by the above equation where [tex]\langle\cdot,\cdot\rangle[/tex] is the canonical inner product induced on p-forms by the Riemannian metric g (it is just the tensor p-product of the inverse metric).
    It is quite easy to derive [tex]d^*=*d*[/tex]. But does anyone know how to do this without using Hodge star operator, through the g-induced inner product directly?
  2. jcsd
  3. Mar 5, 2008 #2
    what text are you referencing?
  4. Mar 6, 2008 #3
    Any good differential geometry book should have this. One I like, which is for physicist's is John Baes' "Knots, Gauge Theory and Gravity". You might also want to check out Bleecker's " Variational Principles in Gauge Theories". I re-read the post and it seemed a bit badly written. So just to spell out what I meant:
    The Hodge star is defined by:
    [tex]\langle \lambda_1,\lambda_2\rangle vol=\lambda_1\wedge*\lambda_2[/tex]
    Find the adjoint of d without using Hodge star, just the canonically induced metric [tex]\langle\cdot,\cdot\rangle[/tex]
  5. Mar 6, 2008 #4
    thanks, Henry.

    I have R.W.R. Darling's "Differential Forms and Connections". Formal. Not written for physicists. And a 1985 dover reprint, Dominic G. B. Edelen, "Applied Exterior Calculus". Neither one have I read yet.

    Perhaps you could tell me if either of these texts might be worth trying.

    I don't want to be misleading. I find differential forms fascinating in their application to physics. Sadly, I'm not capable of responding to your past three posts, as yet, but I would surely like to get to that point. From what I've seen over the past 6 weeks, Hurkyl seems to talk of differential forms with some authority. You might try buttonholing him for some input.

    Last edited: Mar 6, 2008
  6. Mar 8, 2008 #5


    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Silly question -- if you want to express [itex]* d *[/itex] without the Hodge stars, what's wrong with simply replacing them with a formula that calculates them?

    Or... maybe your question is more fundamental? You call it the adjoint, so I assume

    [tex]\langle d^* f, g \rangle = \langle f, dg \rangle?[/tex]

    Was that what you wanted? Or maybe something like [itex]d^*f[/itex] is the transpose of the tangent multi-vector [itex]\langle f, d \_\_\_ \rangle[/itex]?
    Last edited: Mar 8, 2008
  7. Mar 9, 2008 #6
    Because it is not really the answer I am looking for, but to apply a different method to obtain adjoints when we do not have the Hodge star. Thanks for the answer!
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook