Just as for functions $f_1,f_2 \in L^2(\Omega_1)$, with $\Omega_1 \subset \mathbb R ^d$ and the Euclidean metric, we have: $$\langle f_1, f_2 \rangle = \int_{\Omega_1} \overline{f_1 (x)} f_2(x) dx$$
What is the analog inner product on a manifold $(\mathcal M, g)$? Can it be expressed as this?:
$$\langle f_1, f_2 \rangle_g = \int_\mathcal{M} \overline{f_1(x)} f_2(x) g^*(x) dx$$
for some function $g^*(x)$ determined by $g$?
I am rather unfamiliar with basic concepts of Differential Geometry, but I am familiar with introductory Functional Analysis. Most of the descriptions I have found of this involve notation/technical details I am unfamiliar with.