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

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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.

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  • $\begingroup$ Thank you for this reference. We can define the tangent space in the usual sense i presume? $\endgroup$
    – Netivolu
    Apr 18 '18 at 11:40
  • $\begingroup$ @Netivolu I am not sure what you mean by a tangent space. I am not sure if $L^2(C,M)$ has a Banach manifold structure. $\endgroup$ Apr 18 '18 at 12:35
  • $\begingroup$ Well i wondered if you could construct a manifold (Hilbert, Banach, Frechet) structure on those $L^2$ spaces $\endgroup$
    – Netivolu
    Apr 18 '18 at 13:01
  • $\begingroup$ @Netivolu I honestly don't know. The problem is that since the mappings are not necessarily continuous, you cannot localize them in a coordinate chart in the image. $\endgroup$ Apr 18 '18 at 19:23

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