📂Geometry

Differentiable Manifolds

Definition¹

Let $M$ be an arbitrary set and $U_{\alpha} \subset \mathbb{R}^{n}$ be an open set. For the function $1-1$ $\mathbf{x}_{\alpha} : U_{\alpha} \to M$, the ordered pair $\left( M, \left\{ \mathbf{x}_{\alpha} \right\}_{\alpha\in \mathscr{A}} \right)$, or simply $M$, is defined as a differentiable manifold of dimension $n$ if the following conditions are met:

1. $\bigcup \limits_{\alpha} \mathbf{x}_{\alpha} \left( U_{\alpha} \right) = M$
2. For $\varnothing \ne W = \mathbf{x}_{\alpha}\left( U_{\alpha} \right) \cap \mathbf{x}_{\beta}\left( U_{\beta} \right)$, the mapping $\mathbf{x}_{\beta}^{-1} \circ \mathbf{x}_{\alpha} : \mathbf{x}_{\alpha}^{-1}(W) \to \mathbf{x}_{\beta}^{-1}(W)$ is differentiable.
3. For all possible $\alpha$ that satisfy conditions 1 and 2, the index family $\left\{ \left( U_{\alpha}, \mathbf{x}_{\alpha} \right) \right\}$ is constructed.

Description

• Also simply called a differentiable manifold or smooth manifold. A $n$-dimensional differential manifold is sometimes denoted by $M^{n}$.

• When $p \in \mathbf{x}_{\alpha}(U_{\alpha})$, $\left( U_{\alpha}, \mathbf{x}_{\alpha} \right)$ or simply $\mathbf{x}_{\alpha}$ is called the coordinate system of $M$ at $p$, local coordinate system, or parameterization.

• The $\mathbf{x}_{\alpha}(U_{\alpha})$ is called the coordinate neighborhood at $p \in M$.

• The index family $\left\{ \left( U_{\alpha}, \mathbf{x}_{\alpha} \right) \right\}$ satisfying condition 3. is called the differentiable structure on $M$

• For $p \in M$, functions $x_{i}$ that satisfy $\mathbf{x}_{\alpha}^{-1}(p) = \left( x_{1}(p), \dots, x_{n}(p) \right)$ are called coordinate functions.

1. Because $M$ is given as a completely arbitrary set (i.e., not generally a metric space), there can be no discussion about whether $\mathbf{x}_{\alpha}$ is differentiable or not. Moreover, since $M$ is a union of various images, a suitably good condition is needed at each intersection $W = \mathbf{x}_{\alpha}\left( U_{\alpha} \right) \cap \mathbf{x}_{\beta}\left( U_{\beta} \right)$, which is given here as the condition of being differentiable.

   Depending on the condition of mapping $\mathbf{x}_{\beta}^{-1} \circ \mathbf{x}_{\alpha}$, the manifold is called various names. For instance, if the condition of being continuous is given instead of differentiable, then $M$ becomes a topological manifold. If the condition of being holomorphic is given, then $M$ becomes a complex manifold. Also, if $\mathbf{x}_{\beta}^{-1} \circ \mathbf{x}_{\alpha} \in C^{k}$, then $M$ is called a $C^{k}$ manifold. In differential geometry, the tool of differentiation is used to describe geometry, hence the handling of differentiable manifolds.

2. This content is a technical part, existing to avoid discussions about whether two differentiable structures are the same or different. Assuming that all such satisfying 1 and 2 have been gathered, it means, ‘How about this?’, ‘Is this also included?’ such tackles should not be thrown.

Example

Euclidean Space $\mathbb{R}^{n}$

$$ \mathbb{R}^{n} = \left\{ (x_{1}, x_{2}, \dots, x_{n}) : x_{i} \in \mathbb{R} \right\} $$

It’s natural to consider $\mathbb{R}^{n}$ as a differentiable manifold because a manifold is locally similar to Euclidean space. Let’s call ${\rm id}$ the identity operator.

1. The differentiable structure is established as $\left\{ \left( U_{\alpha}, {\rm id} \right) | U_{\alpha} \subset \mathbb{R}^{n} \text{ is open.} \right\}$.

2. Since the identity operator is differentiable, it is established.

3. For all such pairs, the index family $\left\{ \left( U_{\alpha}, {\rm id} \right)\right\}$ is constructed.

Thus, $\left( \mathbb{R}^{n}, \left\{ {\rm id} \right\} \right)$ is a differentiable manifold.

2-dimensional Sphere $\mathbb{S}^{2}$

$$ \mathbb{S}^{2} = \left\{ p \in \mathbb{R}^{3} : \left\| p \right\|=1 \right\} $$

A 2-dimensional sphere can be represented with 6 coordinate patches as follows. For $(u,v) \in U = \left\{ (u,v) : u^{2} + v^{2} \lt 1 \right\}$,

Coordinate Patch	Definition	Inverse
$\mathbf{x}_{1} = \mathbf{x}_{(0,0,1)} : U \to \R^{3}$	$\mathbf{x}_{(0,0,1)}(u, v) = \left( u, v , \sqrt{1- u^{2} -v^{2} } \right)$	$\mathbf{x}_{(0,0,1)}^{-1}(x, y, z) = (x,y)$
$\mathbf{x}_{2} = \mathbf{x}_{(0,0,-1)} : U \to \R^{3}$	$\mathbf{x}_{(0,0,-1)}(u, v) = \left( u, v , -\sqrt{1- u^{2} -v^{2} } \right)$	$\mathbf{x}_{(0,0,-1)}^{-1}(x, y, z) = (x,y)$
$\mathbf{x}_{3} = \mathbf{x}_{(0,1,0)} : U \to \R^{3}$	$\mathbf{x}_{(0,1,0)}(u, v) = \left( u, \sqrt{1- u^{2} -v^{2}}, v \right)$	$\mathbf{x}_{(0,1,0)}^{-1}(x, y, z) = (x,z)$
$\mathbf{x}_{4} = \mathbf{x}_{(0,-1,0)} : U \to \R^{3}$	$\mathbf{x}_{(0,-1,0)}(u, v) = \left( u, -\sqrt{1- u^{2} -v^{2}}, v \right)$	$\mathbf{x}_{(0,-1,0)}^{-1}(x, y, z) = (x,z)$
$\mathbf{x}_{5} = \mathbf{x}_{(1,0,0)} : U \to \R^{3}$	$\mathbf{x}_{(1,0,0)}(u, v) = \left( \sqrt{1- u^{2} -v^{2}}, u, v \right)$	$\mathbf{x}_{(1,0,0)}^{-1}(x, y, z) = (y,z)$
$\mathbf{x}_{6} = \mathbf{x}_{(-1,0,0)} : U \to \R^{3}$	$\mathbf{x}_{(-1,0,0)}(u, v) = \left( -\sqrt{1- u^{2} -v^{2}}, u, v \right)$	$\mathbf{x}_{(-1,0,0)}^{-1}(x, y, z) = (y,z)$

1. $\bigcup \limits_{i=1}^6 \mathbf{x}_{i} = \mathbb{S}^{2}$ is established.

2. $\mathbf{x}_{(0,0,1)}^{-1} \circ \mathbf{x}_{(1,0,0)}$, being as follows, is differentiable.

$$\mathbf{x}_{(0,0,1)}^{-1} \circ \mathbf{x}_{(1,0,0)}(u,v) = \mathbf{x}_{(0,0,1)}^{-1} \left( \sqrt{1- u^{2} -v^{2}}, u, v \right) = \left( \sqrt{1- u^{2} -v^{2}}, u \right) \in C^{\infty}$$

3. In this manner, for all pairs satisfying 1 and 2, the index family $\left\{ \left( U_{\alpha}, \mathbf{x}_{\alpha} \right) \right\}$ is constructed.

Thus, $\left( \mathbb{S}^{2} , \left\{ \mathbf{x}_{\alpha} \right\} \right)$ is a differentiable manifold.