defintion (smooth manifold): A smooth manifold M is a topological manifold with an maximal atlas of compatible charts $(U_i, \psi_i)_{i \in I}$.
definition (submanifold): A $n$-dimensional submanifold is a subset $\tilde{M} \subset M$ of a smooth $k$-dimensional manifold M such that for each $p \in \tilde{M}$ there exist a chart $(U,\psi)$ of $M$ with $p \in U$ and $$\psi(\tilde{M} \cap U)=V \cap (\mathbb{R}^n \cap \{0\}) \subset \mathbb{R}^k$$
($V=\psi(U)$).
question: Does the chart $(U,\psi)$ has to be an element of the maximal atlas or just any chart of $M$? And if it could be any chart, does then even exist for each $p \in \tilde{M}$ a chart in the maximal atlas with the properties above?