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Let $M$ be a smooth manifold of dimension $n$. A subset $S$ of $M$ is said to be a regular submanifold of $M$ of dimension $k$ if, for every point $p \in S$, there exists a coordinate chart $(U,\phi)$ of $M$ around $p$ such that $S \cap U$ is given by the vanishing of (last) $n-k$ coordinates.

In this case, we call $(U,\phi)$ an adapted coordinate chart around $p$ relative to $S$. I am curious about the terminology "adapted coordinate chart". Is it because $S$ has "adapted" the coordinate chart of $M$ in some sense?

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Your problem is one of language, not of mathematics. In particular, your sentence

Is it because $S$ has "adapted" the coordinate of $M$ in some sense?

suggests that you're mistaking the word "adapted" for "adopted". As far as I can tell, the terminology simply comes from the fact that the chart on $M$ is a somewhat special chart, adapted to $S$ in order to given a nice form to $S\cap U$.

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In general a coordinate system of the ambient manifold has no relation to a given submanifold. The term adapted just means that the coordinate system has a form in which the submanifold can be identified in an easy way.

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