Hodge star

<math>\pi_1(S^1)=?\,</math>

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

Let <math>M</math> be a Riemannian manifold in <math>n</math> dimensions with metric <math>g</math>. The Hodge star operator is a linear operator from i-differential forms to (n-i)-differential forms

<math>

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

</math>

defined as follows: Let <math>\phi_1,...,\phi_n</math> be a local orthonormal coframe (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

<math>

• \phi_1\wedge\cdots\wedge\phi_i = \pm \phi_{i+1}\wedge\cdots\wedge\phi_n

</math>

where the plus or minus is chosen so that

<math>\phi_1\wedge\cdots\wedge\phi_i\wedge *(\phi_1\wedge\cdots\wedge\phi_i)</math>

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

Example

Give <math>R^2</math> the standard metric so that <math>dx, dy</math> is a coframe. Then the volume form is <math>dx\wedge dy</math>. Thus,

<math>

• dx = dy

</math>

and

<math>

• dy = -dx

</math>

and in general

<math>

• fdx + gdy = fdy - gdx

</math>