Musical Isomorphisms

Question

I'm studying from Fecko's Differential Geometry and Lie Groups for Physicists, and in the part introducing metric tensors, Fecko introduces the musical isomorphisms between the tangent and cotangent space as the following maps for some metric tensor $g$. $$\flat{g}:v \mapsto g(v, \cdot)$$ and similarly, $$\sharp{g}:\alpha \mapsto g^{-1}(\alpha, \cdot)$$ Fecko then gives an exercise to show these equal, in component form, the following: $$g_{ab}v^b$$ and $$g^{ab}\alpha_b$$ respectively. How do I calculate $g(v, \cdot)$ or $g^{-1}(\alpha, \cdot)$ ? I'm not sure where to begin... I'm new to tensors and get lost easily with the notation.

Answer 1

If the components of the metric tensor $g$ are $g_{ab} = g(e_a,e_b)$ and $v= \sum v^a e_a$ then $g(v, \cdot)$ is a 1-form, which in the dual basis $\{f_k\}$ its $c$-component is $g(v^a e_a, e_c)= v^a g_{ac}$.

Considering the inverse metric, whose components are $g^{m n}$ in the above dual basis, you can obtain in a similar way the second expression.