# Definition A set $M$ is defined to be a *manifold* if each point of *M* has an [[Open Set|open]] [[neighborhood]] which has a [[continuous map|continuous]] [[Types of Maps|bijection]] onto an [[open set]] of $\mathbb{R}^n$ ([[Real Coordinate Space|R^n]]) for some $n$. The [[dimension]] of the manifold is then $n$. More simply, it means that $M$ locally "looks like" $\mathbb{R}^n$. The definition has to involve only open sets and not the whole of $M$ and $\mathbb{R}^n$ because we do not want to affect the global topology of $M$. Moreover, the [[map]] is only required to be [[Types of Maps|bijective]], not preserve any geometrical notions. Given a point $P$ on a manifold $M$, the aforementioned bijection assigns to $P$ an $n$-tuple of numbers $(x^1(P), x^2(P), \ldots x^n(P)) = x^i(P)$. These numbers are called the *coordinates* of $P$ under the map. See the figure below. ![[Pasted image 20210123175907.png]] Another way of thinking about an $n$-dimensional manifold is as a set which can be given $n$-independent coordinates in some neighborhood of any point, since these coordinates actually define the bijection discussed above. ^e43e20 Because every manifold is locally the same as some $\mathbb{R}^n$, any two manifolds of the same dimension and [[Differentiable Manifold|differentiability class]] are locally indistinguishable. Thus, manifolds are divided into classes depending on their global structure. # Examples The concept of a manifold embraces sets which one might not ordinarily regard as spaces. By definition, *any* set $M$ that can be parameterized continuously is a manifold whose dimension is the number of independent parameters. For example: 1. The set of all rotations of a rigid object in 3D is a manifold, since it can be parameterized by three "Euler angles". This is a Lie group known as [[SO(3)]]. 2. [[Lie Group|Lie groups]] more generally are defined manifolds. 3. For $N$ particles, the numbers consisting of all their positions ($3N$ numbers) and velocities ($3N$ numbers) define a point in a $6N$ dimensional manifold, the [[Phase Space]]. ## Vector spaces as manifolds [[Vector Space|Vector spaces]] are commonly encountered and are actually manifolds. We can construct a [[map]] from a vector space to some $R^n$ ([[Real Coordinate Space|R^n]]). Supposed that $V$ is n-[[Dimension|dimensional]], and choose any [[basis]] $\{\pmb{e}_i\}$. We can write any $\pmb{y} \in V$ as $\pmb{y} = \sum_i a^i \pmb{e}_i$. However, since $\pmb{y}$ is a point in $V$, this establishes a relationship from $V$ to $\mathbb{R}^n$, $\pmb{y} \mapsto a^i$. In fact, every point in $\mathbb{R}^n$ corresponds to a unique vector in $V$ under this map, so $V$ is covered by a single [[Coordinate Chart|chart]] and isomorphic to $\mathbb{R}^n$. Thus, we can think of any vector space $V$ as $\mathbb{R}^n$ when convenient to do so.