Let $M$ be a Riemannian manifold, and let $p \in M$.
I know that the tangent space at $p$, $T_pM$, is isomorphic as a vector space to the cotangent space at $p$, $(T_pM)^*$. Thus, in some sense, as vector spaces, they are indistinguishable.
Since $M$ is a Riemannian manifold, $T_pM$ has an inner product, the Riemannian metric, and thus is not just a vector space, but also a Hilbert space. Thus elements $v \in T_pM$ have notions of angle and length, and can be thought of as "geometric vectors", i.e. arrows pointing from some origin.
Do elements $w \in (T_pM)^*$ also have notions of length and angle, allowing them to be thought of as geometric vectors? Or is $(T_pM)^*$ just a metrizable topological vector space, allowing one to think of $w \in (T_pM)^*$ as "points", objects for which there is a meaningful notion of distance and a "Cartesian grid", as well as notions of addition and scalar multiplication, but not of length or angle.
Overarching question: do the tangent space at $p$, $T_pM$, and the cotangent space at $p$, $(T_pM)^*$, have any different structures defined on them which are not defined on their counterpart? And for those structures which are defined on both sets, for which are they isomorphic?
- Is the cotangent space also a Hilbert space? If so, are the tangent and cotangent spaces isomorphic as Hilbert spaces?
- Is the cotangent space also a Banach space? If so, are they isomorphic as Banach spaces?
- Are they both metric spaces? If so, are they isomorphic as metric spaces?