In Baez & Munian's book Gauge Fields, Knots, and Gravity, when introducing the Hodge star operator, they say
At any point $p$ in a 3-dimensional Riemannian manifold $M$, the Hodge star operator maps a 1-form $\nu$, which we draw as a little arrow, into a 2-form $\omega \wedge \mu$ that corresponds to an area element that is orthogonal to $\nu$.
Conversely, it maps $\omega \wedge \mu$ to $\nu$. In general, in $n$ dimensions the Hodge star operator maps $p$-forms to $(n-p)$-forms in a very similar way, taking each '$p$-dimensional area element' to an orthogonal '$(n-p)$-dimensional area element'.
I am unclear on what they mean by orthogonality here, I assume they mean the inner product $\langle \nu, \omega \wedge \mu \rangle = 0$. Earlier they defined what it means to take the inner product of two differential forms of the same degree, but in the above passage they are different degrees. What does it mean for a 1-form to be orthogonal to a 2-form?
Best Answer
Suppose $\{e_1,\dots, e_n\}$ is an orthonormal basis for an $n$-dimensional (pseudo) inner product space $(V,g)$, and let $*$ be the Hodge star on the exterior powers of $V$. Then for any increasing sequence of indices $I=\{i_1,\dots, i_k\}$, let $I^c= \{1,\dots, n\}\setminus I =\{j_1,\dots, j_{n-k}\}$ be the remaining set of indices. Then, $\star e_I=\pm e_{I^c}$, i.e \begin{align} \star (e_{i_1}\wedge\cdots\wedge e_{i_k})=\pm e_{j_1}\wedge \cdots\wedge e_{j_{n-k}}. \end{align} How do we think of the wedge product $e_{i_1}\wedge\cdots\wedge e_{i_k}$? Well, it is common to think of this as the subspace $E_I=\text{span}\{e_{i_1}\dots, e_{i_k}\}$. So, the Hodge star takes a $k$-vector (intuitively we imagine it as a $k$-dimensional subspace spanned by those vectors) and it maps it to an $(n-k)$-vector, and by the formula above (which follows in a few lines from the definitions) this $(n-k)$-vector is such that the associated subspace $E_{I^c}$ is the orthogonal complement to the one we started from.
If you don’t wish to think of the entire subspace, another common interpretation is that it is the ‘unit box’ spanned by these vectors, and once again, we see that a $k$-dimensional box is mapped to an orthogonal $(n-k)$ box.
They are not talking about inner products between different types of objects.