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I am familiarizing myself with Riemannian manifolds.

Let $M$ be an orientable smooth $n$-manifold with atlas $(U_\alpha, (x_1^\alpha, \dots, x_n^\alpha))$ and let $g$ be a Riemannian metric on $M$. Let $|g_\alpha|$ denote the determinant of the matrix given by $$ (g_\alpha)_{ij} = g\left ( {\partial \over \partial x_i^\alpha} , {\partial \over \partial x_j^\alpha}\right )$$

I want to show that $\sqrt{|g_\alpha|} d x_1^\alpha \wedge \dots \wedge d x_n^\alpha = \sqrt{|g_\beta|} d x_1^\beta \wedge \dots \wedge d x_n^\beta$ on $U_\alpha \cap U_\beta$ but I can't quite do it.

Could someone help me how to show $\sqrt{|g_\alpha|} d x_1^\alpha \wedge \dots \wedge d x_n^\alpha = \sqrt{|g_\beta|} d x_1^\beta \wedge \dots \wedge d x_n^\beta$?

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learner - here's one approach.

If $\phi: U \rightarrow V$ is a change of coordinates, say $U$ has coordinates $x_i$ and $V$ has coordinates $y_i$, and the metric is written in $V$ as $(g_{ij})$, and in $U$ as $(h_{ij})$, then the metric transforms as $$(*) \quad \quad (h_{ij}) = d\phi^T (g_{ij}) d\phi$$.

So, the top form on $U$ defined on terms of $(h_{ij})$ is $\lvert \det(h_{ij}) \rvert^{1/2} dx_i$.

On the other hand, the top form on $V$ is similarly $\lvert \det(g_{ij}) \rvert^{1/2} dy_i$, and this pulls back to $\lvert \det(g_{ij}) \rvert^{1/2} \det d\phi dy_i$. Using (*) and asking that your change of coordinates is oriented gives the desired result.

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  • $\begingroup$ Thank you. What is a "top form"? I have never seen the term before. I understand $d\phi$ is a vector in the basis ${\partial \over \partial x^i}$. Right? $\endgroup$ – self-learner Jan 26 '15 at 10:01
  • $\begingroup$ Hi self-learner, you are welcome. Please feel free to ignore'top form', I just meant that if we on an $n$ dimensional manifold, all $n+1, n+2, ...$ differential forms are 0, so an expression $f dx_1 \wedge \ldots dx_n$ is `top' in the sense it has the maximal amount of wedges. By $d \phi$ I meant the Jacobian matrix of $\phi$, so yes $d\phi$ converts expressions of the form $a_i \frac{\partial}{\partial y_i }$ into $b_j \frac{\partial}{\partial x_j}$, i.e. is the infinitesimal change of coordinates. $\endgroup$ – uncookedfalcon Jan 27 '15 at 23:57

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