5.5 Integration of differential forms

5.5.1 Orientation

An integration of a differential form over a manifold $M$ is defined only when $M$ is ‘orientable’. So we first define an orientation of a manifold. Let $M$ be a connected $m$ -dimensional differentiable manifold. At a point $p\in M$ , the tangent space $T_{p}M$ is spanned by the basis $\{e_{\mu }\}=\{\partial /\partial x^{\mu }\}$ , where $x^{\mu }$ is the local coordinate on the chart $U_i$ to which $p$ belongs. Let $U_j$ be another chart such that $U_i\cap U_j\ne \varnothing$ with the local coordinates $y^{\alpha }$ . If $p\in U_i\cap U_j$ , $T_{p}M$ is spanned by either $\{e_{\mu }\}$ or $\{{\tilde{e}}_{\alpha }\}=\{\partial /\partial y^{\alpha }\}$ . The basis changes as

$${\tilde{e}}_{\alpha }=\left(\frac{\partial x^{\mu }}{\partial y^{\alpha }}\right)e_{\mu }.\text{\hspace{1em}(5.97)}$$

If $J=\det (\partial x^{\mu }/\partial y^{\alpha })>0$ on $U_i\cap U_j$ , $\{e_{\mu }\}$ and $\{{\tilde{e}}_{\alpha }\}$ are said to define the same orientation on $U_i\cap U_j$ and if $J<0$ , they define the opposite orientation.

Definition 5.6

Let $M$ be a connected manifold covered by $\{U_i\}$ . The manifold $M$ is orientable if, for any overlapping charts $U_i$ and $U_j$ , there exist local coordinates $\{x^{\mu }\}$ for $U_i$ and $\{y^{\alpha }\}$ for $U_j$ such that $J=\det (\partial x^{\mu }/\partial y^{\alpha })>0$ .

If $M$ is non-orientable, $J$ cannot be positive in all intersections of charts. For example, the Möbius strip in figure 5.14(a) is non-orientable since we have to choose $J$ to be negative in the intersection B.

(a)

(b)

Figure 5.14. (a) The Möbius strip is obtained by twisting the part ${\mathbf{B}}^{\prime }$ of the second strip by $\pi$ before pasting A with ${\mathbf{A}}^{\prime }$ and B with ${\mathbf{B}}^{\prime }$ . The coordinate change on B is $y^1=x^1$ , $y^2=-x^2$ and the Jacobian is $-1$ . (b) Basis frames on the Möbius strip.

If an $m$ -dimensional manifold $M$ is orientable, there exists an $m$ -form $\omega$ which vanishes nowhere. This $m$ -form $\omega$ is called a volume element, which plays the role of a measure when we integrate a function $f\in \mathcal{F}(M)$ over $M$ . Two volume elements $\omega$ and ${\omega }^{\prime }$ are said to be equivalent if there exists a strictly positive function $h\in \mathcal{F}(M)$ such that $\omega =h{\omega }^{\prime }$ . A negative-definite function $h^{\prime }\in \mathcal{F}(M)$ gives an inequivalent orientation to $M$ . Thus, any orientable manifold admits two inequivalent orientations, one of which is called right handed, the other left handed. Take an $m$ -form

$$\omega =h(p)\mathrm{d}{x}^1\wedge \cdots \wedge \mathrm{d}{x}^m\text{\hspace{1em}(5.98)}$$

with a positive-definite $h(p)$ on a chart $(U,\phi )$ whose coordinate is $x=\phi (p)$ . If $M$ is orientable, we may extend $\omega$ throughout $M$ such that the component $h$ is positive definite on any chart $U_i$ . If $M$ is orientable, this $\omega$ is a volume element. Note that this positivity of $h$ is independent of the choice of coordinates. In fact, let $p\in U_i\cap U_j\ne \varnothing$ and let $x^{\mu }$ and $y^{\alpha }$ be the coordinates of $U_i$ and $U_j$ , respectively. Then (5.98) becomes

$$\omega =h(p)\frac{\partial x^1}{\partial y^{{\mu }_1}}\mathrm{d}{y}^{{\mu }_1}\wedge \cdots \wedge \frac{\partial x^m}{\partial y^{{\mu }_m}}\mathrm{d}{y}^{{\mu }_m}=h(p)\det \left(\frac{\partial x^{\mu }}{\partial y^{\nu }}\right)\mathrm{d}{y}^1\wedge \cdots \wedge \mathrm{d}{y}^m.\text{\hspace{1em}(5.99)}$$

The determinant in (5.99) is the Jacobian of the coordinate transformation and must be positive by assumed orientability. If $M$ is non-orientable, $\omega$ with a positive-definite component cannot be defined on $M$ . Let us look at figure 5.14 again. If we circumnavigate the strip along the direction shown in the figure, $\omega =\mathrm{d}x\wedge \mathrm{d}y$ changes the signature $\mathrm{d}x\wedge \mathrm{d}y\to -\mathrm{d}x\wedge \mathrm{d}y$ when we come back to the starting point. Hence, $\omega$ cannot be defined uniquely on $M$ .

5.5.2 Integration of forms

Now we are ready to define an integration of a function $f:M\to \mathbb{R}$ over an orientable manifold $M$ . Take a volume element $\omega$ . In a coordinate neighbourhood $U_i$ with the coordinate $x$ , we define the integration of an $m$ -form $f\omega$ by

$${\int }_{U_i}f\omega \equiv {\int }_{\phi (U_i)}f\left({\phi }_i^{-1}(x)\right)h\left({\phi }_i^{-1}(x)\right)d{x}^1\dots d{x}^m.\text{\hspace{1em}(5.100)}$$

The RHS is an ordinary multiple integration of a function of $m$ variables. Once the integral of $f$ over $U_i$ is defined, the integral of $f$ over the whole of $M$ is given with the help of the ‘partition of unity’ defined now.

Definition 5.7

Take an open covering $\{U_i\}$ of $M$ such that each point of $M$ is covered with a finite number of $U_i$ . [If this is always possible, $M$ is called paracompact, which we assume to be the case.] If a family of differentiable functions ${\epsilon }_i(p)$ satisfies

(i) $0\le {\epsilon }_i(p)\le 1$
(ii) ${\epsilon }_i(p)=0$ if $p\notin U_i$ and
(iii) ${\epsilon }_1(p)+{\epsilon }_2(p)+\dots =1$ for any point $p\in M$

the family $\{\epsilon (p)\}$ is called a partition of unity subordinate to the covering $\{U_i\}$ .

From condition (iii), it follows that

$$f(p)=\sum _{i}f(p){\epsilon }_i(p)=\sum _i{f}_i(p)\text{\hspace{1em}(5.101)}$$

where $f_i(p)\equiv f(p){\epsilon }_i(p)$ vanishes outside $U_i$ by (ii). Hence, given a point $p\in M$ , assumed paracompactness ensures that there are only finite terms in the summation over $i$ in (5.101). For each $f_i(p)$ , we may define the integral over $U_i$ according to (5.100). Finally the integral of $f$ on $M$ is given by

$${\int }_{M}f\omega \equiv \sum _i{\int }_{U_i}{f}_i\omega .\text{\hspace{1em}(5.102)}$$

Although a different atlas $\{(V_i,{\psi }_i)\}$ gives different coordinates and a different partition of unity, the integral defined by (5.102) remains the same.

Example 5.13

Let us take the atlas of $S^1$ defined in example 5.2. Let $U_1=S^1-\{(1,0)\}$ , $U_2=S^1-\{(-1,0)\}$ , ${\epsilon }_1(\theta )={\sin }^2(\theta /2)$ and ${\epsilon }_2(\theta )={\cos }^2(\theta /2)$ . The reader should verify that $\{{\epsilon }_i(\theta )\}$ is a partition of unity subordinate to $\{U_i\}$ . Let us integrate a function $f={\cos }^2\theta$ , for example. [Of course we know

$${\int }_0^{2\pi }d\theta {\cos }^2\theta =\pi$$

but let us use the partition of unity.] We have

$$\begin{array}{rl}{\int }_{S^1}\mathrm{d}\theta {\cos }^2\theta & ={\int }_0^{2\pi }\mathrm{d}\theta {\sin }^2\frac{\theta }{2}{\cos }^2\theta +{\int }_{-\pi }^{\pi }\mathrm{d}\theta {\cos }^2\frac{\theta }{2}{\cos }^2\theta \\ & =\frac{1}{2}\pi +\frac{1}{2}\pi =\pi .\end{array}$$

So far, we have left $h$ arbitrary provided it is strictly positive. The reader might be tempted to choose $h$ to be unity. However, as we found in (5.99), $h$ is multiplied by the Jacobian under the change of coordinates and there is no canonical way to single out the component $h$ ; unity in one coordinate might not be unity in the other. The situation changes if the manifold is endowed with a metric, as we will see in chapter 7.

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正文：英文 · OCR 机器稿 · 待校对

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