In a Riemannian manifold, its inner product:
$\left( \cdot,\cdot \right)_g : T_pM \times T_pM \to K$
Can be defined as by pairing $\alpha^i$ with $\beta^j$ dual by means of the metric tensor:
$\left( \alpha,\beta \right)_g= \left< \alpha^i, g_{ij}\beta^j \right>$
But also via Hodge-star operator:
$\left( \alpha,\beta \right)_g= \alpha^i \wedge \star\beta^j$
From my (may be wrong) understanding, these definitions look very different to me. One arise only from metric concepts, while the second is about "completing" the space.
Question: Is it just that we equip our same $T_pM$ with two different inner products or are they just equivalent definitions? If they are, how are they related then?
Thank you