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Chapter 8: Problem 13

Suppose \(M\) is a compact orientable \(n\)-manifold (with no boundary), and \(\theta\) is an \((n-1)\)-form on \(M\). Show that \(d \theta\) is 0 at some point.

Short Answer

Expert verified
By Stokes' theorem and properties of integrals, \(d\theta\) is zero at least at one point.

Step by step solution

01

Understand the Given Problem

We need to show that the exterior derivative of an \(n-1\)-form \(\theta\) on a compact orientable \(n\)-manifold \(M\) is zero at least at one point.
02

Recall Stokes' Theorem

Stokes' Theorem states that for a manifold \(M\) with boundary \(\partial M\), and an \(n-1\)-form \(\theta\), \(\int_{\partial M} \theta = \int_{M} d\theta\). Since \(M\) has no boundary, \(\partial M = \emptyset\), and hence \(\int_{\partial M} \theta = 0\).
03

Application of Stokes' Theorem to our Manifold

Since \(\partial M = \emptyset\), Stokes' theorem tells us \(\int_{M} d\theta = 0\).
04

Integrals of Non-Negative Functions

The integral of a function over a compact manifold is zero only if the function itself is zero at some point. If \(d \theta\) were non-zero everywhere, the integral \(\int_{M} d\theta\) could not be zero.
05

Conclusion

Thus, since \(\int_{M} d\theta = 0\), it must be that \(d \theta\) is zero at some point in the manifold \(M\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Stokes' theorem
Stokes' theorem is a powerful tool in differential geometry that relates the integral of a differential form over a manifold to the integral of its exterior derivative over the boundary of that manifold. Mathematically, it is often stated as follows:
For an \(n-1\)-form \(\theta\) on an \(n\)-manifold \(M\) with boundary 鈭侻: \[\int_{\partial M} \theta = \int_{M} d\theta.\]
However, if the manifold \(M\) has no boundary (i.e., \(\partial M = \emptyset\)), the left side of the theorem becomes zero, leading to:
\[\int_{M} d\theta = 0.\] This result can be extremely useful, as it simplifies the study of differential forms on such manifolds and provides profound insights into their structure and properties.
Exterior derivative
The exterior derivative is an operation that takes a differential form and returns a new form of higher degree. If \(\theta\) is an \(n-1\)-form, then \(d\theta\) is an \(n\)-form. This operator is linear and satisfies two crucial properties:
* Linearity: \(d(a\theta + b\psi) = ad\theta + bd\psi\) where \(a\) and \(b\) are constants.
* The property \(d \circ d = 0\) means that applying the exterior derivative twice is always zero, i.e., if \(\theta\) is a form, then \(d(d\theta) = 0\).
In the context of our problem, if \(\theta\) is an \(n-1\)-form on a compact orientable \(n\)-manifold without boundary, then Stokes' theorem tells us \[\int_{M} d\theta = 0.\]This condition implies that \(d\theta\) must be zero at some point on the manifold.
Manifold without boundary
A manifold without boundary is a topological space that locally resembles Euclidean space and has no edge or boundary. This means any small neighborhood within the manifold looks like a piece of \(\mathbb{R}^n\).
When working with compact orientable manifolds without boundary in differential geometry, we encounter interesting theorems like Stokes' theorem which simplify greatly because terms involving the boundary disappear (since there is no boundary).
In our problem, the concept of a manifold without boundary is crucial because it leads directly to the simplification \(\partial M = \emptyset\), which then allows us to leverage Stokes' theorem to argue that \[\int_{M} d\theta = 0.\]Geez Consequently, the exterior derivative \(d\theta\) cannot be non-zero everywhere, meaning it must be zero at some point on \(M\).

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Most popular questions from this chapter

Let \(f: M^{n} \rightarrow N^{n}\) be a proper map between oriented \(n\)-manifolds such that \(f_{*}: M_{p} \rightarrow N_{f(\rho)}\) is orientation preserving whenever \(p\) is a regular point. Show that if \(N\) is connected, then either \(f\) is onto \(N\), or else all points are critical points of \(f\).

(a) Let \(M^{n}\) and \(N^{m}\) be oriented manifolds, and let \(\omega\) and \(\eta\) be an \(n\)-form and an \(m\)-form with compact support, on \(M\) and \(N\), respecuively. We will orient \(M \times N\) by agreeing that \(v_{1}, \ldots, v_{n}, w_{1}, \ldots, w_{m}\) is positively oriented in \((M \times N)_{(p, q)} \approx M_{p} \oplus N_{q}\) if \(v_{1}, \ldots, v_{n}\) and \(w_{1}, \ldots, w_{m}\) are positively oriented in \(M_{p}\) and \(N_{q}\), respectively. If \(\pi_{i}: M \times N \rightarrow M\) or \(N\) is projection on the \(i^{\text {th }}\) factor, show that $$ \int_{M \times N} \pi_{1}^{*} \omega \wedge \pi_{2}{ }^{*} \eta=\int_{M} \omega \cdot \int_{N} \eta $$ (b) If \(h: M \times N \rightarrow \mathbb{R}\) is \(C^{\infty}\), then $$ \int_{M \times N} h \pi_{1}^{*} \omega \wedge \pi_{2}^{*} \eta=\int_{M} g \omega $$ where $$ g(p)=\int_{N} h(p, \cdot) \eta, \quad h(p, \cdot)=q \mapsto h(p, q) $$ (c) Every \((m+n)\)-form on \(M \times N\) is \(h \pi_{1}{ }^{*} \omega \wedge \pi_{2}^{*} \eta\) for some \(\omega\) and \(\eta\).

For two maps \(f, g: M \rightarrow N\) we will write \(f \simeq g\) to indicate that \(f\) is smoothly homotopic to \(g\). (a) If \(f \simeq g\), then there is a smooth homotopy \(H^{\prime}: M \times[0,1] \rightarrow N\) such that $$ \begin{aligned} &H^{\prime}\langle p, t)=f(p) \quad \text { for } t \text { in a neighborhood of } 0 \\ &H^{\prime}(p, t)=g(p) \quad \text { for } t \text { in a neighborhood of } 1 \end{aligned} $$ (b) \(\simeq\) is an equivalence relation.

Let \(M\) and \(N\) be compact \(n\)-manifolds, and let \(f, g: M \rightarrow N\) be smoothly homotopic, by a smooth homotopy \(H: M \times[0,1] \rightarrow N\). (a) Let \(q \in N\) be a regular value of \(H\). Let # \(f^{-1}(q)\) denote the (finite) number of points in \(f^{-1}(q)\). Show that $$ \\# f^{-1}(q) \equiv \\# g^{-1}(q) \quad(\bmod 2) $$ Hint: \(H^{-1}(q)\) is a compact I-manifold-with-boundary. The number of points in its boundary is clearly even. (This is one place where we use the stronger form of Sard's Theorem.) (b) Show, more gencrally, that this result holds so long as \(q\) is a regular value of both \(f\) and \(g\).

Let \(\left\\{X^{\prime}\right\\}\) be a \(C^{\infty}\) family of \(C^{\infty}\) vector fields on a compact manifold \(M .\) (To be more precisc, suppose \(X\) is a \(C^{\infty}\) vector ficld on \(M \times[0,1] ;\) then \(X^{\prime}(p)\) will denote \(\left.\pi_{M *} X_{(p, t) \cdot}\right)\) From the addendum to Chapter 5, and the argument which was used in the proof of Theorem \(5-6\), it follows that there is a \(C^{\infty}\) family \(\left\\{\phi_{t}\right\\}\) of diffeomorphisms of \(M\) [not necessarily a ]-parameter group], with \(\phi_{0}=\) identity, which is generated by \(\left\\{X^{t}\right\\}\), i.e., for any \(C^{\infty}\) function \(f: M \rightarrow \mathbb{R}\) we have $$ \left\langle X^{\prime} f\right)(p)=\lim _{h \rightarrow 0} \frac{f\left(\phi_{t+h}(p)\right)-f\left(\phi_{t}(p)\right)}{h} $$ For a family \(\omega_{1}\) of \(k\)-forms on \(M\) we define the \(k\)-form $$ \dot{\omega}_{t}=\lim _{h \rightarrow 0} \frac{\omega_{t+h}-\omega_{t}}{h} $$ (a) Show that for \(\eta\langle t)=\phi_{1}{ }^{*} \omega_{t}\) we have $$ \dot{\eta}_{t}=\phi_{t}{ }^{*}\left(L_{X^{\prime}} \omega_{t}+\dot{\omega}_{t}\right) $$ (b) Let \(\omega_{0}\) and \(\omega_{\mathrm{I}}\) be nowhere zero \(n\)-forms on a compact oriented \(n\)-manifold \(M\), and define $$ \omega_{t}=(1-t) \omega_{0}+t \omega_{1} $$ Show that the family \(\phi_{t}\) of diffeomorphisms generated by \(\left\\{X^{\prime}\right\\}\) satisfies $$ \phi_{t}^{*} \omega_{t}=\omega_{0} \quad \text { for all } t $$ if and only if $$ L_{X^{t}} \omega_{t}=\omega_{0}-\omega_{1} $$ (c) Using Problem 7-18, show that this holds if and only if $$ \left.d\left(X^{t}\right\lrcorner \omega_{t}\right)=\omega_{0}-\omega_{1} $$ (d) Suppose that \(\int_{M} \omega_{0}=f_{M} \omega_{1}\), so that \(\omega_{0}-\omega_{1}=d \lambda\) for some \(\lambda\). Show that there is a diffeomorphism \(f_{1}: M \rightarrow M\) such that \(\omega_{0}=f_{1}{ }^{*} \omega_{1}\).
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