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Chapter 2: Problem 17

Show that the set of phase curves on the surface \(\pi_{E_{0}}\) forms a twodimensional sphere. The formula \(w=\left(x_{1}+i y_{1}\right) /\left(x_{2}+i y_{2}\right)\) gives the "Hopf map" from the three sphere \(\pi_{E_{0}}\) to the two sphere (the complex \(w\)-plane completed by the point at infinity). Our phase curves are the pre-images of points under the Hopf map.

Short Answer

Expert verified
Question: Show that the set of phase curves on the surface \(\pi_{E_{0}}\) forms a two-dimensional sphere. Answer: The set of phase curves on the surface \(\pi_{E_{0}}\) forms a two-dimensional sphere because the pre-images of points under the Hopf map connect to form a closed surface. This is demonstrated by analyzing the pre-images of the points and showing that as we move along the two-dimensional sphere, the radius of the pre-images changes while their shapes still form circles. Therefore, by connecting all these circles, a two-dimensional sphere is formed.

Step by step solution

01

Understanding the Hopf map

The Hopf map is a map from the three-dimensional sphere \(\pi_{E_{0}}\) to the two-dimensional sphere (the complex w-plane completed by the point at infinity). It is defined as: $$ w = \frac{x_1 + iy_1}{x_2 + iy_2} $$ The Hopf map is used to project points from the three-dimensional sphere onto the two-dimensional sphere.
02

Find the Pre-images of Points

The pre-images of points under the Hopf map are sets of points in the three-dimensional sphere that are mapped onto the same point in the two-dimensional sphere. To find the pre-images of a point \(w\), we need to solve the equation and find all \((x_1, y_1, x_2, y_2)\) in the three-dimensional sphere that satisfy \(w=\frac{x_1 + iy_1}{x_2 + iy_2}\): $$ w(x_2 + iy_2) = x_1 + iy_1 $$ Multiply both sides of the equation by the complex conjugate \((x_2 - iy_2)\) in the denominator: $$ w(x_2^2 + y_2^2) = (x_1 + iy_1)(x_2 - iy_2) $$ This equation represents the pre-images of the point \(w\) under the Hopf map.
03

Analyze the Pre-images

Now, let's analyze the pre-images of the point \(w\) under the Hopf map. Since \(x_1^2 + y_1^2 + x_2^2 + y_2^2 = E_0^2\) (the radius of \(\pi_{E_{0}}\)), we have: $$ w^2(x_2^2 + y_2^2) = (x_1^2 + y_1^2) $$ This equation tells us that the set of pre-images for each point \(w\) lies on a circle and the radii of these circles are related by a factor of \(w^2\). Therefore, as we move along the two-dimensional sphere, the radius of the pre-images will change, but their shapes will still form a circle.
04

Combining the Pre-images

Each point on the two-dimensional sphere corresponds to a certain circle on the three-dimensional sphere \(\pi_{E_{0}}\). We have to find out the structure of all pre-images together. The phase curves will form a two-dimensional sphere if we can connect all these circles and form a closed surface. Since moving around the two-dimensional sphere results in moving around circles in the \(\pi_{E_{0}}\), and any two points on the two-dimensional sphere can be connected by a path in the \(\pi_{E_{0}}\), it can be concluded that the set of phase curves indeed forms a two-dimensional sphere. Therefore, we have shown that the set of phase curves on the surface \(\pi_{E_{0}}\) forms a two-dimensional sphere.

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