Compatability of Coordinates

Compatibility of Coordinates

Statements

Definition. The coordinate neighborhoods $(U,\phi )$ and $(V,\psi )$ are said to be ${\mathcal{C}}^r-$compatible if $U\cap V\ne \varnothing$ implies that there exists $r\in {\mathbb{N}}_{>0}$ such that, for indices $i,j\in {\mathbb{N}}_{>0}$, the functions $h^i$ and $g^j$ giving the change of coordinates are of class ${\mathcal{C}}^r$. This is equivalent to requiring $\phi \circ {\psi }^{-1}$ and $\psi \circ {\phi }^{-1}$ to be diffeomorphisms of the open subsets $\phi (U\cap V)$ and $\psi (U\cap V)$ of ${\mathbb{R}}^n$.

References

@Boo75 Chapter 3, definition 1.1