= Inner product on exterior powers of a cotangent space
{title2=$\langle\xi_1\wedge\cdots\wedge\xi_p,\zeta_1\wedge\cdots\wedge\zeta_p\rangle=\det(\langle\xi_i,\zeta_j\rangle)$}
A <Riemannian metric> induces a dual <inner product> on each <cotangent space>. On decomposable covectors define $\langle\xi_1\wedge\cdots\wedge\xi_p,\zeta_1\wedge\cdots\wedge\zeta_p\rangle=\det(\langle\xi_i,\zeta_j\rangle)$ and extend bilinearly. Increasing wedges of an orthonormal coframe form an orthonormal basis. This positive-definite inner product is the one used to define the <Hodge star operator>.
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