Given a compact manifold $C$ and a n-manifold $M$ we mostly work on either
$C^{\infty}(C,M)$ seen as a Frechet manifold.
or $H^{k}(C,M)$ seen has a Hilbert manifold when $k > n/2$.
Although both spaces have advantages i am interested to know if we can extend the construction given in the second case to define $H^0(C,M)$.
Or more in general when can we define the space $L^2(C,M)$ outside of the known examples where $M = \mathbb{R}^n$?
Assuming that $M$ and $N$ are Riemannian manifolds, the space $L^p(M,N)$ consists of measurable mappings $f:M\to N$ such that $x\to d(y_0,f(x))$ belongs to $L^p(M)$. There is no problem with this definition if the measure of $M$ is finite and a small problem if the measure of $M$ is infinite. Indeed, in the later case the constant mapping $f(x)=y_0$ belongs to $L^p$ because $d(y_0,f)\equiv 0$, but if we change the point $y_0$ to $y_1\neq y_0$, then $d(y_1,f)=d(y_1,y_0)\neq 0$ is not in $L^p$.
Actually, one can define $L^p(X,Y)$ mappings from a measure space $X$ into a separable metric space as follows: There is an isometric embedding of $Y$ into $\ell^\infty$ and then we define: $$ L^p(X,Y)=\{f\in L^p(X,\ell^\infty):\, f(x)\in Y \text{ a.e.}\}. $$ Here the the $L^p$ integral of $\ell^\infty$ valued mappings is understood as the Bochner integral.
For more details see for example:
J. Heinonen, P. Koskela, N. Shanmugalingam, J. T. Tyson, Sobolev spaces on metric measure spaces. An approach based on upper gradients. New Mathematical Monographs, 27. Cambridge University Press, Cambridge, 2015.
