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Let $\phi:\mathbb{R}^n\to\mathbb{R}^n$ be an orthogonal linear map. Prove that $\phi^*(*\alpha) = *\phi^*(\alpha)$ for all $k$-forms $\alpha$ on $\mathbb{R}^n$.

I tried to write out $\phi^*(*\alpha)$ and $*\phi^*(\alpha)$, but I don't see where linearity and orthogonality comes into the proof. Any ideas?

QD666
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HINT: Let's write $\phi\colon V\to W$, both $V$ and $W$ being $\Bbb R^n$. If $\alpha_i$ give an orthonormal basis for $W^*$, let $\beta_i = \phi^*\alpha_i$ and show that these give an orthonormal basis for $V^*$. It suffices to consider $\alpha = \alpha_{i_1}\wedge\dots\wedge\alpha_{i_k}$. What is $\star\alpha$? Now express $\phi^*(\alpha)$ and $\phi^*(\star\alpha)$ in terms of the $\beta_i$'s.

Ted Shifrin
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  • Oh, sorry. What is $W^*$? – QD666 Feb 11 '19 at 02:08
  • The dual space of $W$, i.e., the space of linear functionals on $W$. That's where $dx_1, \dots, dx_n$ live. – Ted Shifrin Feb 12 '19 at 07:11
  • thank you so much – QD666 Feb 12 '19 at 16:02
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    @TedShifrin: does Hodge star commute with pull back? If so co-differential also commute with pull back ($\delta=(-1)^{n(k+1)+1}\star d\star$). Therefore pull back preserves harmonic forms?! – C.F.G Apr 02 '20 at 09:02
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    Not in general, @C.F.G. Note that in this question we had an orthogonal linear map. So, same dimensions and metric-preserving. – Ted Shifrin Apr 02 '20 at 16:25
  • In general the Hodge star commutes with pullback by orientation preserving isometries; see https://math.stackexchange.com/questions/267654/isometry-and-harmonic-forms – DavideL May 12 '23 at 09:54