inner product, hodge dual def

[Orbots]

This is might help resolve tension between vee and inner product.
NecroMonster — Yesterday at 6:22 AM
however, a "dot product" is essentially an operation mapping a vector to a covector (the musical isomorphism defined by the metric) and the contraction with another vector

[Orbots]

hamish_todd — Yesterday at 6:29 AM
But if you're doing differential geometry and you're using vectors and covectors then you almost certainly have a metric?
NecroMonster — Yesterday at 6:31 AM
if you want to have a levi-civita connection? yeah
but you dont have to
differential geometry is doable without a metric structure