Hodge star

$\pi _{1}(S^{1})=?\,$

This article/section deals with mathematical concepts appropriate for a student in late university or graduate level.

Let $M$ be a Riemannian n-manifold with metric $g$. The Hodge star operator is a linear operator from i-differential forms to (n-i)-differential forms

$*:\Omega ^{i}(T^{*}M)\rightarrow \Omega ^{n-i}(T^{*}M)$

defined as follows: Let $\phi _{1},...,\phi _{n}$ be a local orthonormal co-frame (i.e., a collection of locally defined 1-forms which are orthonormal with respect to the induced metric on the cotangent space). Then we define

$*\phi _{1}\wedge \cdots \wedge \phi _{i}=\pm \phi _{i+1}\wedge \cdots \wedge \phi _{n}$

where the plus or minus is chosen so that

$\phi _{1}\wedge \cdots \wedge \phi _{i}\wedge *(\phi _{1}\wedge \cdots \wedge \phi _{i})$

is the volume form on $M$. To define the Hodge star operator for general forms, we simply extend the above definition by linearity.

Example

Give $R^{2}$ the standard metric so that $dx,dy$ is a coframe. Then the volume form is $dx\wedge dy$. Thus,

$*dx=dy$

and

$*dy=-dx$

and in general

$*fdx+gdy=fdy-gdx$