15.1 Differentiable manifolds

中文

A locally Euclidean space or topological manifold $M$ of dimension $n$ is a Hausdorff topological space $M$ in which every point $x$ has a neighbourhood homeomorphic to an open subset of ${\mathbb{R}}^n$ . If $p$ is any point of $M$ then a (coordinate) chart at $p$ is a pair $(U,\varphi )$ where $U$ is an open subset of $M,$ , called the domain ofthe chart and $\varphi :U\to \varphi (U)\subset {\mathbb{R}}^n$ is a homeomorphism between $U$ and its image $\varphi (U)$ . The image $\varphi (U)$ is an open subset of ${\mathbb{R}}^n$ , given the relative topology in ${\mathbb{R}}^n$ . It is also common to call $U$ a coordinate neighbourhood of $\dot{\mathbf{p}}$ and $\varphi$ a coordinate map. The functions $x^i={\mathrm{pr}}_i\circ \varphi :U\mathbb{R}(i=1,\dots ,n)$ , where ${\mathfrak{pr}}_i:{\mathbb{R}}^n\to R$ are the standard projection maps, are known as the coordinate functions determined by this chart, and the real numbers $x^i(p)$ are called the coordinates of $p$ in this chart (see Fig. 15.1). Sometimes, when we wish to emphasize the symbols to be used fo the coordinate functions, we denote the chart by $(U,\varphi ;x^i)$ , or simply $(U;x^i)$ . Occasionally the term coordinate system at $p$ is used for a chart whose domain $U$ covers $p.$ . The use of superscripts rather than subscripts for coordinate functions is not universal, but its advantages will become apparent as the tensor formalism on manifolds is developed.

For any pair ofcoordinate charts $(U,\varphi ;x^i)$ ) and $(U^{\prime },{\varphi }^{\prime };x^{\prime j})$ such that $U\cap U^{\prime }\ne \varnothing .$ , define the transition functions

$$\begin{array}{l}{\varphi }^{\prime }\circ {\varphi }^{-1}:\varphi (U\cap U^{\prime })\to {\varphi }^{\prime }(U\cap U^{\prime }),\\ \varphi \circ {\varphi }^{-1}:{\varphi }^{\prime }(U\cap U^{\prime })\to \varphi (U\cap U^{\prime }),\end{array}$$

Figure 15.1 Chart at a point $p$

Figure 15.2 Transition functions on compatible chart

which are depicted in Fig. 15.2. The transition functions are often written

$$x^{\prime j}=x^{\prime j}(x^1,x^2,\dots ,x^n)\text{ and }{x}^i=x^i(x^{\prime 1},x^{\prime 2},\dots ,x^{\prime n})(i,j=1,\dots ,n)\text{\hspace{1em}(15.1)}$$

which is an abbreviated form of the awkward, but technically correct,

$$\begin{array}{c}{x}^{\prime j}(p)={\mathsf{pr}}_i\circ {\varphi }^{\prime }\circ {\varphi }^{-1}(x^1(p),x^2(p),\dots ,x^n(p)),\\ x^i(p)={\mathsf{pr}}_i\circ \varphi \circ {\varphi }^{\prime -1}(x^{\prime 1}(p),x^{\prime 2}(p),\dots ,x^{\prime n}(p)).\end{array}$$

The two charts are said to be $C^r$ -compatible where $r$ is a non-negative integer or $\infty ,$ if al the functions in Eq. (15.1) are $C^r$ . For convenience we will generally assume that the charts are $C^{\infty }-compatible.$

An atlas on $M$ is a family of charts $\mathcal{A}=\{(U_{\alpha },{\varphi }_{\alpha })|\alpha \in A\}$ such that the coordinate neighbourhoods $U_{\alpha }$ cover $M,$ , and any pair of charts from the family are $C^{\infty }$ -compatible. If $\mathbf{A}$ and ${\mathcal{A}}^{\prime }$ are two atlases on $M$ then so is their union $\mathcal{A}\cup \mathcal{A}$

Exercise

Prove this statement. [Hint: A differentiable function of a differentiable function is always differentiable.]

Any atlas $\mathbf{A}$ may thus be extended to a maximal atlas by adding to it all charts that are $C^{\infty }$ -compatible with the charts of . This maximal atlas is called a differentiable structure on $M$. A pair $(M,A)$ , where $M$, is an $n$-dimensional topological manifold and is a differentiable structure on $M,$ , is called a differentiable manifold; it is usually just denoted $M$.

The Jacobian matrix $\mathbf{J}=[\partial x^{\prime k}/\partial x^j]$ is non-singular since its inverse is ${\mathsf{J}}^{-1}=[\partial x^i/\partial x^{\prime k}]$

$${\mathsf{J}}^{-1}\mathsf{J}=\left[\frac{\partial x^i}{\partial x^{\prime k}}\frac{\partial x^{\prime k}}{\partial x^j}\right]=\left[\frac{\partial x^i}{\partial x^j}\right]=\left[{\delta }_j^i\right]=\mathsf{I}.$$

Similarly $\mathsf{J}{\mathsf{J}}^{-1}=\mathsf{I}.$ . Hence the Jacobian determinant is non-vanishing, $det$ $[\partial x^{\prime j}/\partial x^i]\ne 0$ We are making a return here and in the rest of this book to the summation convention of earlier chapters.

Example 15.1

Euclidean space ${\mathbb{R}}^n$ is trivially a manifold, since the single chart $(U={\mathbb{R}}^n,\varphi =\mathrm{id})$ covers it and generates a unique atlas consisting ofall charts that are compatible with it. For example, in ${\mathbb{R}}^2$ it is permissible to use polar coordinates $(r,\theta )$ defined by

$$x=r\cos \theta ,y=r\sin \theta ,$$

which are compatible with $(x,y)$ on the open set $U={\mathbb{R}}^2-\{(x,y)|x\ge 0,y=0\}$ . The inverse transformation is

$$r=\sqrt{x^2+y^2},\theta =\{\begin{array}{ll}\arctan y/x& \text{ if }y>0,\\ \pi & \text{ if }y=0,x<0,\\ \pi +\arctan y/x& \text{ if }y<0.\end{array}$$

The image set $\varphi (U)$ in the $(r,\theta )-\mathsf{plane}$ is a semi-infinite open strip $r>0,0<\theta <2\pi$

Example 15.2

Any open region $U$ of ${\mathbb{R}}^n$ is a differentiable manifold formed by giving it the relative topology and the differentiable structure generated by the single chart $(U,{\mathrm{id}}_{U}U\to {\mathbb{R}}^n)$ . Every chart on $U$ is the restriction ofa coordinate neighbourhood and coordinate map on ${\mathbb{R}}^n$ to the open region $U$ and can be written $(U\cap V,\psi {|}_{U\cap V})$ where $(V,\psi )$ is a chart on ${\mathbb{R}}^n$ . Such a manifold is called an open submanifold of ${\mathbb{R}}^n$

Exercise

Describe the open region of ${\mathbb{R}}^3$ and the image set in the $(r,\theta ,\varphi )$ on which spherical pola coordinates are defined,

$$x=r\sin \theta \cos \varphi ,y=r\sin \theta \sin \varphi ,z=r\cos \theta .\text{\hspace{1em}(15.2)}$$

Example 15.3

The unit circle $S^1\subset {\mathbb{R}}^2$ , defined by the equation $x^2+y^2=1$ , is a onedimensional manifold. The coordinate $x$ can be used on either the upper semicircle $y>0$ or the lower semicircle $y<0_{:}$ , but not on all of $S^1$ . Alternatively, setting $r=1$ in polar coordinates as defined in Example 15.1, a possible chart is $(U,\varphi ;\theta )$ where $U=S^1-\{(1,0)\}$ and $\varphi :U\to \mathbb{R}$ is defined by $\varphi ((x,y))=\theta$ . The image set $\varphi (U)$ is the open interva $(0,2\pi )\subset \mathbb{R}$ . These charts are clearly compatible with each other. $S^1$ is the only one dimensional manifold that is not homeomorphic to the real line $\mathbb{R}$

Example 15.4

The 2-sphere $S^2$ defined as the subset of points $(x,y,z)$ of ${\mathbb{R}}^3$ satisfying

$$x^2+y^2+z^2=1$$

is a two-dimensional differentiable manifold. Some possible charts on $S^2$ are:

(i) Rectangular coordinates $(x,y)$ , defined on the upper and lower hemisphere, $z>0$ and $z<0$ , separately. These two charts are non-intersecting and do not cover the sphere since points on the central plane $z=0$ are omitted.

(ii) Stereographic projection from the north pole, Eqs. (10.1) and (10.2), defines a chart $(S^2-\{(0,0,1)\},{\mathrm{St}}_N)$ where ${\mathrm{St}}_N:(x,y,z)\mapsto (X,Y)$ is given by

$$X=\frac{x}{1-z},Y=\frac{y}{1-z}.$$

These coordinates are not defined on the sphere’s north pole $N=(0,0,1)$ ), but a similar projection ${\mathsf{St}}_S$ from the south pole $S=(0,0,-1)$ will cover $N$,

$$X^{\prime }=\frac{x}{1+z},Y^{\prime }=\frac{y}{1+z}.$$

Both of these charts are evidently compatible with the rectangular coordinate charts (i) and therefore with each other in their region of overlap.

(iii) Spherical polar coordinates $(\theta ,\varphi )$ defined by seting $r=1$ in Eq. (15.2). Simple algebra shows that these are related to the stereographic coordinates (ii) by

$$X=\cot \frac{1}{2}\theta \cos \varphi ,Y=\cot \frac{1}{2}\theta \sin \varphi ,$$

and therefore form a compatible chart on their region of definition.

In a similar way the $n$-sphere $S^n$ ,

$$S^n=\left\{\mathbf{x}\in {\mathbb{R}}^{n+1}|{\left(x^1\right)}^2+{\left(x^2\right)}^2+\cdots +{\left(x^{n+1}\right)}^2=1\right\}$$

is a differentiable manifold of dimension $n$. A set of charts providing an atlas is the set of rectangular coordinates on all hemispheres, $(U_i^{+},{\varphi }_i^{\pm })$ and $(U_i^{-},{\varphi }_i^{\pm })$ ), where

$$U_i^{+}=\{\mathbf{x}\in S^n\mid x^i>0\},U_i^{-}=\{\mathbf{x}\in S^n\mid x^i<0\},$$

and ${\varphi }_i^{+}:U_i^{+}\to {\mathbb{R}}^n$ and ${\varphi }_i^{-}:U_i^{-}\to {\mathbb{R}}^n$ are both defined by

$${\varphi }_i^{\pm }(\mathbf{x})=\left(x^1,x^2,\dots ,x^{i-1},x^{i+1},\dots ,x^{n+1}\right).$$

Exercise

Prove that ${\mathrm{St}}_N$ and ${\mathrm{St}}_S$ are compatible, by showing they are related by

$$X^{\prime }=\frac{X}{X^2+Y^2},Y^{\prime }=\frac{Y}{X^2+Y^2}.$$

Example 15.5

The set of $n\times n$ real matrices $M(n,\mathbb{R})$ can be put in one-to-one correspondence with points of ${\mathbb{R}}^{n^2}$ , through the map $\varphi :M(n,\mathbb{R})\to {\mathbb{R}}^{n^2}$ defined by

$$\varphi (A=[a_{ij}])=\left(a_{11},a_{12},\dots ,a_{1n},a_{21},a_{22},\dots ,a_{nn}\right).$$

This provides $M(n,\mathbb{R})$ with a Hausdorff topology inherited from $R^{n^2}$ in the obvious way. The differentiable structure generated by the chart $(M(n,\mathbb{R}),\varphi )$ converts $M(n,\mathbb{R})$ into a differentiable manifold of dimension $n^2$

The group of $n\times n$ real non-singular matrices $GL(n,\mathbb{R})$ consists of $n\times n$ real matrices having non-zero determinant. The determinant map $det$ : $GL(n,\mathbb{R})\to \mathbb{R}$ is continuous since it is made up purely ofpolynomial operations, so that $\varphi (GL(n,\mathbb{R}))=\det {\mathring{A}}^{-1}(\mathbb{R}-\{0\})$ is an open subset of ${\mathbb{R}}^{n^2}$ . Thus $GL(n,\mathbb{R})$ is a differentiable manifold of dimension $n^2$ , as it is in one-to-one correspondence with an open submanifold of ${\mathbb{R}}^{n^2}$

From any two differentiable manifolds $M$ and $N$ of dimensions $m$ and $n$ respectively, it is possible to form their product $M\times N$ , which is the topological space defined in Section 10.4. Let $(U_{\alpha },{\varphi }_{\alpha })$ and $(V_{\beta },{\psi }_{\beta })$ be any families of mutually compatible charts on $M$ and

$N$ respectively, which generate the differentiable structures on these manifolds. The charts $(U_{\alpha }\times V_{\beta },{\varphi }_{\alpha }\times {\psi }_{\beta })$ , where ${\varphi }_{\alpha }\times {\psi }_{\beta }:U_{\alpha }\times V_{\beta }\to {\mathbb{R}}^m\times {\mathbb{R}}^n={\mathbb{R}}^{m+n}$ defined by

$${\varphi }_{\alpha }\times {\psi }_{\beta }((p,q))=({\varphi }_{\alpha }(p),{\psi }_{\beta }(q))=(x^1(p),\dots ,x^m(p),y^1(q),\dots ,y^n(q)),$$

manifestly cover $M\times N$ , and are clearly compatible in their overlaps. The maximal atlas generated by these charts is a differentiable structure on $M\times N$ making it into a differen tiable manifold of dimension $m+n$

Example 15.6

The topological 2-torus $T^2=S^1\times S^1$ (see Example 10.13) can be given a differentiable structure as a product manifold in the obvious way from the manifold structure on $S^1$ . Similarly, one can define the $n$-torus to be the product of $n$ circles, $T^n=S^1\times S^1\times \cdots \times S^1={\left(S^1\right)}^n$

Problems

Problem 15.1

Show that the group of unimodular matrices $SL(n,\mathbb{R})=\{A\in GL(n,\mathbb{R})|$ $det$ $A=1\}$ is a differentiable manifold.

Problem 15.2

On the $n$-sphere $S^n$ find coordinates corresponding to (i) stereographic projection, (ii) spherical polars.

Problem 15.3

Show that the real projective $n$-space $P^n$ defined in Example 10.15 as the set of straight lines through the origin in ${\mathbb{R}}^{n+1}$ is a differentiable manifold of dimension $n$, by finding an atlas of compatible charts that cover it.

Problem 15.4

Define the complex projective $n$-space $C{P}^n$ in a similar way to Example 10.15 as lines in ${\mathbb{C}}^{n+1}$ ofthe form $\lambda (z^0,z^1,\dots ,z^n)$ where $\lambda ,z^0,\dots ,z^n\in \mathbb{C}$ . Show that $C{P}^n$ is a differentiable (real) manifold of dimension 2$n$.