Discrete Contraction and Inner Product

Let \(\alpha\) and \(\beta\) be two 1-forms. Their inner product can be expressed as

\[(\alpha,\beta)=\langle\alpha^{\sharp},\beta\rangle=\langle\beta^{\sharp},\alpha\rangle=(\alpha^{\sharp},\beta^{\sharp})\]

where \(\langle\cdot,\cdot\rangle\) represents the pairing between a vector field and a one form. In this context one can think of vector fields as row matrices, and forms as row matrices.

The sharp operator is a mapping from forms to vector fields, and the flat operator is its inverse.

\[\begin{split}\begin{aligned} \flat: & \quad & TM\to T^{*}M\\ \sharp: & \quad & T^{*}M\to TM\end{aligned}\end{split}\]

Contraction with a one form

The contraction with a one form is simply given by

\[\langle\alpha^{\sharp},\beta\rangle=(\alpha,\beta)=\star(\alpha\wedge\star\beta)\]

Contraction with a two form

Suppose the two form can be expressed as the wedge product of two one forms such as \(\beta\wedge\gamma\). Then

\[\begin{split}\begin{aligned} \langle\alpha^{\sharp},\beta\wedge\gamma\rangle & = & \langle\alpha^{\sharp},\beta\rangle\wedge\gamma-\langle\alpha^{\sharp},\gamma\rangle\wedge\beta\\ & = & \star(\alpha\wedge\star\beta)\wedge\gamma-\star(\alpha\wedge\star\gamma)\wedge\beta\end{aligned}\end{split}\]