Hodge star

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Let $\text{[math]}$ be a Riemannian n-manifold with metric $\text{[math]}$ . The Hodge star operator is a linear operator from i-differential forms to (n-i)-differential forms

$\text{[math]}$

defined as follows: Let $\text{[math]}$ 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

$\text{[math]}$

where the plus or minus is chosen so that

$\text{[math]}$

is the volume form on $\text{[math]}$ . To define the Hodge star operator for general forms, we simply extend the above definition by linearity.

Example

Give $\text{[math]}$ the standard metric so that $\text{[math]}$ is a coframe. Then the volume form is $\text{[math]}$ . Thus,

$\text{[math]}$

and in general

$\text{[math]}$

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