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 $M = \mathbb{R}^n$?



Does their exist something like L^2 Mapping spaces to general manifolds?
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.


