Differential neighbourhood

of order $2k$ of a manifold $M$ of class $C^p$, $p\geq k+1$

A local chart of the $k$-fold tangent bundle $T^k(M)$.

Comments

For the notion of differential neighbourhood of order $2k$ of a point on a curve see Local differential geometry.

For $k$-fold tangent bundle (also called $k$-th tangent bundle) see Tangent bundle.