Dependence on the class of differentiability between manifolds and maps

Question

Maybe a silly question, but in some books (like "Differential Geometry - Manifolds, Curves, Surfaces - Gostiaux and Berger"), when differentiable maps of class $C^s$ are defined, we have something like:

"Let $f:M \rightarrow N$ a map between $C^k$ manifolds. We say that $f$ is a $C^s$ map (with $s \leq k$) if (...)"

Why (a priori) it's required that "$s \leq k$" ? I tried to find a pathological case, but I had no progress.

Thanks in advance.

Answer 1

Because one tests differentiability of order $s$ in arbitrary charts, and the fact that we have a $C^k$ manifold means charts are only $C^k$ compatible.

What does it mean to say $f(x,y)=y$ is $C^1$ on the $C^0$ manifold $y=|x|$ in $\Bbb R^2$?