Definition of star operator

Question

As picture below, why the star operator have different form on tangent and cotangent (3.3.4 and 3.3.8) ? Besides, why need Riemannian structure for definit star operator on cotangent ?

Answer 1

The part about vector spaces you have copied sounds misleading as it stands. To define a $*$-operation on a vector space, you need an inner product and an orientation. The rule for computing $*(e_{i_1}\wedge\dots\wedge e_{i_p})$ only applies to the elements in an orthonormal basis of $V$. (Otherwise the operation would not be well defined.)

This also explains why (3.3.8) looks different from (3.3.4). The one-forms $dx^i$ usually are not orthonormal (and cannot be chosen to be orthonormal). Computing the $*$-operator in this basis is tedious in general, but for $*(1)$ it is OK: Given an orthonormal basis $e_i$, $dx^1\wedge\dots\wedge dx^n$ is the volume spanned by the vectors $dx^i$ times $e_1\wedge\dots\wedge e_n$, and this leads to (3.3.8).