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I am going crazy... most books don't cover this and instead assume that the manifold is Riemannian or pseudo-Riemannian and has a metric tensor defined on it. I want a "generalized" hodge star.

I have an orientable smooth manifold, thats IT. I have heard that there is a way to formally create a Hodge star/dual between multivectors/forms using only the volume form (a volume form always exists on an orientable manifold). I have really been struggling to do this.

Two things I have asked myself (though do not lead to solutions)

Firstly, does double-application of this kind of "generalized" hodge dual always reproduce the original multivector? I think that it should.

Secondly, does an application of the Hodge dual to "1" (the zero-vector/function = 1 everywhere on the manifold) always need to produce the volume form?

I am really going crazy... please assist :(