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Extension of Riemannian Metric to Higher Forms

I have no problems with understanding the inner product of 1-forms on a Riemannian manifold. We have a metric tensor, it's inverse defines the metric in the cotangent space.

However I'm not very comfortable with the definition of inner product of $p$-froms with $p>1$. It is defined as

$ \langle x^1 \land x^2 \land \ldots \land x^p, y^1 \land y^2 \land \ldots \land y^p \rangle = \det(\langle x^i y^j \rangle) $

Is there a way to hide the determinant? Or at least to write it as a determinant of a sensible linear operator, not just some matrix.

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    Indeed, that what I was looking for.2012-08-21

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