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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


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    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!
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