^^ [[Schutz - 2 - Differentiable Manifolds and Tensors|2. Differentiable Manifolds and Tensors]] <<[[Schutz - 2.2 - The Sphere as a Manifold|2.2 The Sphere as a Manifold]] | [[Schutz - 2.4 - Global Considerations|2.4 Global Considerations]]>> # Other Examples of Manifolds 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) as we'll soon see. ^fc474b 2. 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 [[Schutz - 1.5 - Linear Algebra#^c98ca7|Vector spaces]] are commonly encountered and are actually manifolds. We can construct a map from a vector space to some $R^n$. Supposed that $V$ is n-[[Schutz - 1.5 - Linear Algebra#^3e9444|dimensional]], and choose any [[Schutz - 1.5 - Linear Algebra#^3e9444|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 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. ## Lie Groups The [[Schutz - 2.3 - Other Examples of Manifolds#^fc474b|first example]] above is an example of a Lie Group > **Lie Group** > A Lie group is a [[Schutz - 1.4 - Group Theory#^426790|group]] which is also a [[Schutz - 2.1 - Definition of a Manifold#^31cc42|manifold]] with [[Schutz - 2.1 - Definition of a Manifold#^ddceb2|differentiability class]] $C^\infty$ , with the restriction that the group operation induces a $C^\infty$ map from the manifold into itself. Pick out any element of the group $a$. This element induces a map of $G$ into itself, taking any element $b \in G$ into $ba \in G$, i.e. $b \mapsto ba$. This map must be $C^\infty$ to fit the definition of the Lie Group above. Concretely, this means that in whatever coordinates used on the manifold $G$, the coordinates of $ba$ must be $C^\infty$ function of those of $b$. See [[Schutz - 2.1 - Definition of a Manifold#^c15345|coordinate transformations]] for a review. The demand for such a map is a compatibility requirement, to ensure that the manifold property is compatible with the group property. $\mathbb{R}^n$ is the simplest Lie group (note that all vector spaces are groups and manifolds, and are thus Lie groups since a $C^\infty$ map exists).