Let \(n, k \in \mathbb {N} \) be natural numbers with \(k\ge 1\). A topological Hausdorff space \(M\), together with an open cover \(M=\bigcup _{i\in I}U_{i}\) and the charts. \[\alpha _{i}\colon U_{i}\longrightarrow V_{i}\,\]
with \(V_{i}\subseteq \mathbb {R} ^{n}\) being open subsets of the \(n\)-dimensional metric space of real numbers such that the transition maps. \[\alpha _{j}\circ (\alpha _{i})^{-1}\colon V_{i}\cap \alpha _{i}(U_{i}\cap U_{j})\longrightarrow V_{j}\cap \alpha _{j}(U_{i}\cap U_{j})\,\]
are \(C^{k}\)-diffeomorphisms for all \(i,j\in I\), is called a \(C^{k}\)-manifold or \(k\) times differentiable manifold of the dimension \(n\).
The set of the charts \((U_{i},\alpha _{i})\), \(i\in I\), is called the \(C^{k}\)-atlas of the manifold.