Is $\omega^2$ nowhere vanishing on the four-sphere?

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In summary, the four-sphere is a four-dimensional hypersphere in four-dimensional Euclidean space defined as the set of all points that are a fixed distance from a central point. A differential form on a manifold is said to be nowhere vanishing if it never equals zero at any point on the manifold. $\omega^2$ is important on the four-sphere because it can be used to define the volume form and is always nowhere vanishing on the four-sphere due to its compact and orientable properties. Some applications of this fact include its use in differential geometry, topology, string theory, and the study of gravitational fields and space-time curvature.
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Euge
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Here is this week's POTW:

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If $\omega$ is a two-form on the four-sphere, is $\omega^2$ (i.e., $\omega \wedge \omega$) nowhere vanishing?

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No one answered this week's problem. You can read my solution below.
The answer is no. Suppose to the contrary that $\omega$ is nowhere vanishing. Then $\omega^2$, being a 4-form on a compact oriented 4-manifold has nonzero integral. The de Rham cohomology groups of the four-sphere are $\Bbb R$ in dimensions zero and four, and zero in every other dimension. Thus, $\omega$ is exact. If $\omega = d\eta$, then $\omega^2 = d(\eta \wedge d\eta)$. So $\omega^2$ is exact; its integral is zero by Stokes's theorem. This is absurd.
 

FAQ: Is $\omega^2$ nowhere vanishing on the four-sphere?

What is the four-sphere?

The four-sphere, denoted as $S^4$, is a four-dimensional hypersphere in four-dimensional Euclidean space. It is defined as the set of all points in four-dimensional space that are a fixed distance from a central point, just like how a sphere in three-dimensional space is defined.

What does it mean for $\omega^2$ to be nowhere vanishing?

A differential form $\omega^2$ on a manifold is said to be nowhere vanishing if it never equals zero at any point on the manifold. In other words, there is no point on the manifold where $\omega^2$ is equal to the zero vector.

Why is $\omega^2$ important on the four-sphere?

The four-sphere is a special manifold in which the differential form $\omega^2$ is particularly important. This is because it can be used to define the volume form on the four-sphere, which is necessary for integrating over the manifold and performing various geometric calculations.

Is $\omega^2$ always nowhere vanishing on the four-sphere?

Yes, $\omega^2$ is always nowhere vanishing on the four-sphere. This is because the four-sphere is a compact and orientable manifold, meaning that it is always possible to choose a differential form that is nowhere vanishing on it.

What are some applications of the fact that $\omega^2$ is nowhere vanishing on the four-sphere?

The fact that $\omega^2$ is nowhere vanishing on the four-sphere has many important applications in mathematics and physics. For example, it is used in the study of differential geometry, topology, and string theory. It also has implications in the study of gravitational fields and the curvature of space-time.

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