How to calculate $*dx^i$ on an oriented Riemannian manifold

Question

Let $M$ be a $d$ dimensional oriented Riemannian manifold, $(x^i)$ an oriented local chart, then we can define a star operator $*:\Omega^p(M)\to\Omega^{d-p}(M)$ by looking at orthonormal frames. But how can we calculate $*dx^i$? Can it be expressed as a function of $g^{jk}$ and $dx^1\wedge\cdots\wedge\widehat{dx^j}\wedge\cdots\wedge dx^d$?

Answer 1

Of course it can.

The Hodge star operator is characterized by$$\alpha\wedge*\beta=\langle\alpha,\beta\rangle\mathrm{vol},$$where $\alpha$ and $\beta$ are differential forms of the same degree. So, for $*dx^i$ you have $$dx^j\wedge* dx^i=g^{ij}\mathrm{vol},\quad j=1,\ldots,n.$$Letting $\omega_{ij}$ denote the coefficient of $dx^1\wedge\ldots\widehat{dx^j}\ldots\wedge dx^n$, the above equality means that (up to sign! But I'm not gonna go into signs...) $$\omega_{ij}dx^1\wedge\ldots\wedge dx^n=g^{ij}\mathrm{vol}=g^{ij}\sqrt{\det(g_{ij})}dx^1\wedge\ldots\wedge dx^n,$$and you have what you need.