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?

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# Hodge operator and adjoints

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