A subset $M \subset \R ^n$ is a smooth manifold of dimension $d$ if for every $x \in M$, there exists a neighborhood $V$ of $x$ in $X$ that is diffeomorphic to an open subset $U$ of $\R ^d$. In this case we say that the set is locally diffeomorphic to $\R ^d$.

A diffeomorphism $\phi : U \to V$ is called a parameterization of the neighborhood of $V$. Its inverse diffeomorphism $\phi ^{-1}$ is called a coordinate system (or system of coordinates) on $V$.

We denote the dimension of a manifold $M$ by $\dim M$.

If $X$ and $Z$ are both manifolds in $R^n$
and $Z \subset X$, then we call $Z$ a
submanifold of $X$.
In particular, $X$ is a submanifold of $R^n$.
Any open set of a manifold $X$ is a
submanifold $X$.^{1}

- Future editions will expand. ↩︎