What is the difference of defining a vector field on a manifold and a $C^{k-1}$ vector field on a $C^{k}$ manifold?

Question

In definition number 6.6 here a $C^{k-1}$ vector field is defined on $C^{k}$ manifold, is this merely to express the dimensionality of a vector field or there is more meaning to $C^{k}$ and $C^{k-1}$?

Answer 1

If $\gamma$ is an integral curve for the $C^{k-1}$ vector field $X$, then $\gamma$ is $C^{k}$ (its derivative is $X,$ which can be differentiated a further $k-1$ times). Thus, for $\gamma$ to exist, you should have a notion of $k$'th derivative.