How can I show that the inverse of the induced metric $h_{\alpha \beta}$ is $h^{\alpha \beta}$? So I was reading through Becker, Becker, Schwarz and there is a line in the second chapter that states that $h^{\alpha \beta} =  (h_{\alpha \beta})^{-1}$ where $h_{\alpha \beta}$ is defined as:
$$h_{\alpha \beta} = \frac{\partial X^{\mu}}{\partial \sigma^{\alpha}} \frac{\partial X^{\nu}}{\partial \sigma^{\beta}} g_{\mu \nu}$$
where $X^{\mu}$ is our coordinates on our spacetime manifold, $\sigma^{\alpha}$ is our coordinates on our worldsheet, and $g_{\mu \nu}$ is our spacetime metric. This seems very natural given that $h_{\alpha \beta}$ is precisely the induced metric on the worldsheet and for metrics on our spacetime $g_{\mu \nu}^{-1} = g^{\mu \nu}$. However, I am having a hard time proving this. Namely, 
$$h_{\alpha \beta}h^{\alpha \gamma} = \frac{\partial X^{\mu}}{\partial \sigma^{\alpha}} \frac{\partial X^{\nu}}{\partial \sigma^{\beta}} g_{\mu \nu}\frac{\partial X^{\mu'}}{\partial \sigma_{\alpha}} \frac{\partial X^{\nu'}}{\partial \sigma_{\gamma}} g_{\mu' \nu'}$$
which doesn't look like it will yield $\delta_{\beta}^{\gamma}$. I have tried fiddling with the algebra with no avail...
 A: As you have stated, $h_{\alpha\beta}$ is the induced metric on the worldsheet, which we obtain by acting with what are analogous to projection operators on the metric.
This is a metric in its own right, and by definition the inverse in the matrix sense is $h^{\alpha\beta}$, such that, in $d$ dimensions, $h^{\alpha\beta}h_{\alpha\beta} = \mathrm{Tr} \, \mathbb I = d$.
It should be noted that $h$ is also often used for the fundamental form in most differential geometry literature which whilst related to the induced metric, carries spacetime indices still. The definition most useful to physicists, but not the most general, is that it is given by,
$$h_{\mu\nu} = g_{\mu\nu} \pm n_\mu n_\nu$$
where $n_\mu$ are the unit normals and the sign depends on if they are spacelike or timelike. This one is not as often used in string theory, but it is in general relativity, and I am pointing it out here so you don't get confused between the two.
A: In differential geometry, $h$ is the pullback of the spacetime metric $g$ along the immersion of the worldsheet. Thus it is automatically a metric and writing this in local coordinates will give the formula you outline.
Like any metric, the composition of the dual metric and the metric itself gives the identity.
I'll write out the algebra explicitly later (once I find my notes).
