Hodge star operator

Summary

The Hodge star operator is a linear map that sends $k$-forms to $(n-k)$-forms. It is defined on an oriented $n$-dimensional inner product space and, pointwise, on an oriented Riemannian manifold. It sends a form to its "orthogonal complement", using the metric and the orientation. With it one defines the codifferential, the Hodge Laplacian and harmonic forms, and it underlies Hodge theory and Poincaré duality for de Rham cohomology.