Chapter 17 · Integration on manifolds

中文

The theory of integration over manifolds is only available for a restricted class known as oriented manifolds. The general theory can be found in [1–11]. An $n$-dimensional differentiable manifold $M$ is called orientable if there exists a differential $n$-form $\omega$ that vanishes at no point $p\in M$ . The $n-\mathrm{form}\omega$ is called a volume element for $M$, and the pair $(M,\omega )$ is an oriented manifold. Since the space ${\left({\Lambda }_n\right)}_p\left(M\right)\equiv {\left({\Lambda }^{\ast n}\right)}_p\left(M\right)$ is one-dimensional at each $p\in M$ , any two volume elements are proportional to each other, ${\omega }^{\prime }=f\omega$ , where $f:M\to \mathbb{R}$ is a non-vanishing smooth function on $M$. If the manifold is a connected topological space it has the same sign everywhere; if $f(p)>0$ for all $p\in M$ , the two $n$-forms $\omega$ and ${\omega }^{\prime }$ are said to assign the same orientation to $M$, otherwise they are oppositely oriented. Referring to Example 8.4, a manifold is orientable ifeach cotangent space $T_p^{\ast }(M)\left(p\in M\right)$ is oriented by assigning a non-zero $n$-form $\Omega ={\omega }_p$ at $p$ and the orientations are assigned in a smooth and continuous way over the manifold.

With respect to a coordinate chart $(U,\varphi ;x^i)$ the volume element $\omega$ can be written

If $(U^{\prime },{\varphi }^{\prime };x^{\prime }{}^{i^{\prime }})$ is a second coordinate chart then, in the overlap region $U\cap U^{\prime }$ ,

where

The sign of the component function $g$ thus remains unchanged if and only if the Jacobian determinant of the coordinate transformation is positive throughout $U\cap V$ , in which case the charts are said to have the same orientation. A differentiable manifold is in fact orientable if and only if there exists an atlas of charts $(U_i,{\varphi }_i)$ covering $M$, such that any two charts $(U_i,{\varphi }_i)$ and $(U_j,{\varphi }_j)$ have the same orientation on their overlap $U_i\cap U_j$ , but the proof requires the concept of a partition of unity.

Problems

Problem 17.1

Show that in spherical polar coordinates

$$\mathrm{d}x\wedge \mathrm{d}y\wedge \mathrm{d}z=r^2\sin \theta \mathrm{d}r\wedge \mathrm{d}\theta \wedge \mathrm{d}\varphi ,$$

and that $a^2$ $\sin$ $\theta$ $d\theta$ $d\varphi$ is a volume element on the 2-sphere $x^2+y^2+z^2=a^2$

Problem 17.2

Show that the 2-sphere $x^2+y^2+z^2=1$ is an orientable manifold.

Contents

• 17.1 Partitions of unity
• 17.2 Integration of n-forms
• 17.3 Stokes’ theorem
• 17.4 Homology and cohomology
• 17.5 The Poincaré lemma