I want to prove that $**\omega=\left(-1\right)^{k\left(n-k\right)}\omega$, where $*$ is the Hodge star operator acting on differential $k$-forms $\omega$ on $\mathbb{R}^n$. Where can I find the proof of this?
[Math] Hodge double star operator
differential-geometry
Best Answer
The proof follows directly from applying the definition of the Hodge star twice. The most annoying thing is that you usually need some identity for the contraction of totally antisymmetric tensors. It's spelled out in Nakahara's "Geometry, Topology and Physics", page 291. He defines the dual as $$ \star \omega = \frac{\sqrt{\vert g \vert}}{r!(m-r)!} \omega_{m_1 \dots m_r} \epsilon^{m_1 \dots m_r}_{\phantom{m_1\dots m_r} n_{r+1} \dots n_m} dx^{n_{r+1}} \wedge \dots \wedge dx^{n_m}. $$ Using this twice you'll need some identity for contracting the $\epsilon$'s, which you can calculate quickly using induction (or you can just guess it). If you don't want to get your hands dirty by shifting indices around, I found another version of the proof in Voisin's "Hodge Theory and Complex Algebraic Geometry, volume 1" around page 120. The key to that is that $\star$ preserves metrics, so $$ \langle \alpha, \beta \rangle \text{vol} = \langle \star \alpha, \star \beta \rangle \text{vol}. $$