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I don't understand this definition of the Sobolev space $H^{r,s}(E)$ for a vector bundle:

Def. The section $s:M \rightarrow E$ is contained in the Sobolev space $H^{r,s}(E)$ if for any bundle atlas with the property that on compact sets all coordinate changes and their derivatives are bounded, and for any bundle chart from that atlas, $\phi: E_U \rightarrow U \times R^n$, we have that $\phi \circ s_{|U}$ is contained in $H^{r,s}(U)$.

I understand that the requirement "on compact sets all coordinate changes and their derivatives are bounded" is for compatibility reasons but I don't know what does $H^{r,s}(U)$ mean.

Would the definition of $L^2$ section be the same except for the requirement on the derivatives to be bounded?

Thanks.

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    @PtF I think it was from Jost's book on differential geometry/Riemannian geometry2015-01-03

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$H^{r,s}(U)$ is the usual Sobolev space on the flat domain $U\subset\mathbb{R}^m$.

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    no I meant to $R^m \times R^n$ since the map the section composed with the trivialization of the vector bundle goes to that space, am I right?2012-08-21