Usually the adjoint to the exterior derivative [tex]d^*[/tex] on a Riemannian manifold is derived using the inner product(adsbygoogle = window.adsbygoogle || []).push({});

[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?

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Hodge operator and adjoints

**Physics Forums | Science Articles, Homework Help, Discussion**