Hodge operator and adjoints

  • Thread starter HenryGomes
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In summary, the adjoint to the exterior derivative on a Riemannian manifold is calculated using the inner product
  • #1
Usually the adjoint to the exterior derivative [tex]d^*[/tex] on a Riemannian manifold is derived using the inner product
[tex]\langle\langle\lambda_1,\lambda_2\rangle\rangle:=\int_M\langle\lambda_1,\lambda_2\rangle\mbox{vol}=\int_M\lambda_1\wedge*\lambda_2[/tex]
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?
 
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  • #2
what text are you referencing?
 
  • #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]
 
  • #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.

-deCraig
 
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  • #5
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]?
 
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  • #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!
 

1. What is the Hodge operator and what is its purpose?

The Hodge operator is a mathematical tool used in differential geometry and differential topology. It maps differential forms to other differential forms, and its purpose is to simplify calculations and proofs in these fields.

2. How is the Hodge operator related to adjoints?

The Hodge operator and adjoints are closely related in that they both involve the concept of duality. The Hodge operator is essentially the adjoint of the exterior derivative operator, and this duality allows for simplification of calculations in differential geometry and topology.

3. Can the Hodge operator be generalized to higher dimensions?

Yes, the Hodge operator can be generalized to higher dimensions. In fact, the Hodge operator is defined for any smooth manifold with a Riemannian metric, regardless of its dimension.

4. What is the relationship between the Hodge operator and differential forms?

The Hodge operator acts on differential forms, mapping them to other forms of a different degree. It can be thought of as a way to "rotate" or "transpose" forms, and it is an important tool in the study of differential forms and their properties.

5. How is the Hodge operator used in applications?

The Hodge operator has many applications in mathematics and physics. It is used in areas such as differential geometry, topology, and algebraic geometry. It also has applications in physics, particularly in the study of fields and their associated differential forms.

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