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Chapter 4: Problem 23

Show that the isometries of the unit sphere $$ S^{2}=\left\\{(x, y, z) \in R^{3} ; x^{2}+y^{2}+z^{2}=1\right\\} $$ are the restrictions to \(S^{2}\) of the linear orthogonal transformations of \(R^{3}\).

Short Answer

Expert verified
Isometries of \( S^2 \) are restrictions of orthogonal transformations of \( \mathbb{R}^3 \).

Step by step solution

01

- Understand the problem

Isometries are transformations that preserve distances. We need to show that any isometry of the unit sphere can be described as a restriction of a linear orthogonal transformation of \( \mathbb{R}^3 \).
02

- Recall properties of isometries

Isometries of \( S^2 \) preserve the spherical distance between any two points on the sphere.
03

- Identify linear orthogonal transformations

Linear orthogonal transformations in \( \mathbb{R}^3 \) are those that preserve the Euclidean distance, and they correspond to rotations and reflections, described by orthogonal matrices \( O(3) \).
04

- Show restriction to \( S^2 \) are isometries

Since orthogonal transformations in \( \mathbb{R}^3 \) preserve the Euclidean and hence the spherical distance, their restrictions to \( S^2 \) will also be isometries of \( S^2 \).
05

- Converse: isometries of \( S^2 \) extends to orthogonal transformations

Any isometry of \( S^2 \) can be extended to an orthogonal transformation of \( \mathbb{R}^3 \) by defining the transformation on vectors not on \( S^2 \) via rotation and reflection symmetries.
06

- Conclude the proof

Therefore, isometries of \( S^2 \) are precisely the restrictions to \( S^2 \) of the linear orthogonal transformations of \( \mathbb{R}^3 \).

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Key Concepts

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

Orthogonal Transformations
Orthogonal transformations are linear maps that preserve the Euclidean distance in a space. In mathematical terms, an orthogonal transformation in \( \mathbb{R}^3 \) maintains the lengths of vectors and the angles between them. These transformations are crucial in geometry and physics for preserving the structure of objects.

Orthogonal transformations are represented by orthogonal matrices, where the columns (or rows) are orthonormal vectors. This means that the matrix multiplied by its transpose results in the identity matrix:
\[ A A^T = I \]
Examples of orthogonal transformations include rotations, which pivot an object around a fixed point without altering its shape, and reflections, which flip an object over a specified plane.
Spherical Distance
Spherical distance is the shortest path between two points on the surface of a sphere, commonly measured along the surface rather than through the sphere's interior. This type of distance is particularly important in fields like astronomy and geography, where measurements are often made on a spherical model of the Earth or celestial bodies.

Mathematically, the spherical distance \( d_s \) between two points on the unit sphere can be calculated using the formula:
\[ d_s = \arccos(\( \text{dot product of normalized vectors} \)) \]
This ensures that the computed distance respects the curvature of the sphere, unlike the straight-line (Euclidean) distance.
Euclidean Distance
Euclidean distance is the straight-line distance between two points in Euclidean space, which is the ordinary space of geometry. It is the most common way to measure distance and can be extended to any number of dimensions.

The formula for Euclidean distance \( d_e \) between two points \((x_1, y_1, z_1) \) and \((x_2, y_2, z_2) \) in \( \mathbb{R}^3 \) is given by:
\[ d_e = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]
Euclidean distance is preserved by orthogonal transformations, which means applying a rotation or reflection to points in space does not change the distances between them.
Rotation and Reflection Symmetries
Transformations like rotations and reflections are known as symmetries, as they preserve the overall shape and structure of objects. In three-dimensional space, these symmetries are often described using orthogonal matrices:

* **Rotations:** These are transformations that include spinning an object about an axis by a certain angle. They preserve distances and angles, making them an example of isometries. Even after rotation, the relative distances between points on an object remain the same.
* **Reflections:** These involve flipping an object across a plane, effectively reversing its orientation. This can also be viewed as a form of isometry, ensuring the object鈥檚 geometry remains unchanged though seen in a mirrored form.
Orthogonal Matrices
Orthogonal matrices are square matrices where the rows and columns are orthonormal vectors. This implies that the transpose of the matrix is also its inverse:
\[ A^T A = I \]
Orthogonal matrices form a group known as the orthogonal group, denoted as \( O(n) \), where \( n \) is the dimension. In our context of \( \mathbb{R}^3 \), we deal with \( O(3) \).

These matrices are pivotal in defining linear transformations that maintain angle and distance properties. Whether through rotation (maintaining orientation) or reflection (changing orientation), orthogonal matrices ensure that the underlying Euclidean distances and angles of objects are preserved.

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Most popular questions from this chapter

We say that a differentiable map \(\varphi: S_{1} \rightarrow S_{2}\) preserves angles when for every \(p \in S_{1}\) and every pair \(v_{1}, v_{2} \in T_{p}\left(S_{1}\right)\) we have $$ \cos \left(v_{1}, v_{2}\right)=\cos \left(d \varphi_{p}\left(v_{1}\right), d \varphi_{p}\left(v_{2}\right)\right) . $$ Prove that \(\varphi\) is locally conformal if and only if it preserves angles.

Consider the torus of revolution generated by rotating the circle $$ (x-a)^{2}+z^{2}=r^{2}, y=0, $$ about the \(z\) axis \((a>r>0)\). The parallels generated by the points \((a+r, 0),(a-r, 0),(a, r)\) are called the maximum parallel, the minimum parallel, and the upper parallel, respectively. Check which of these parallels is a. A geodesic. b. An asymptotic curve. c. A line of curvature.

A diffeomorphism \(\varphi: S_{1} \rightarrow S_{2}\) is said to be a geodesic mapping if for every geodesic \(C \subset S_{1}\) of \(S_{1}\), the regular curve \(\varphi(C) \subset S_{2}\) is a geodesic of \(S_{2}\). If \(U\) is a neighborhood of \(p \in S_{1}\), then \(\varphi: U \rightarrow S_{2}\) is said to be a local geodesic mapping in \(p\) if there exists a neighborhood \(V\) of \(\varphi(p)\) in \(S_{2}\) such that \(\varphi: U \rightarrow V\) is a geodesic mapping. a. Show that if \(\varphi: S_{1} \rightarrow S_{2}\) is both a geodesic and a conformal mapping, then \(\varphi\) is a similarity; that is, $$ \langle v, w\rangle_{p}=\lambda\left\langle d \varphi_{p}(v), d \varphi_{p}(w)\right\rangle_{\varphi(p)}, \quad p \in S_{1}, v, w \in T_{p}\left(S_{1}\right), $$ where \(\lambda\) is constant. b. Let \(S^{2}=\left\\{(x, y, z) \in R^{3} ; x^{2}+y^{2}+z^{2}=1\right\\}\) be the unit sphere, \(S^{-}=\) \(\left\\{(x, y, z) \in S^{2} ; z<0\right\\}\) be its lower hemisphere, and \(P\) be the plane \(z=-1\). Prove that the map (central projection) \(\varphi: S^{-} \rightarrow P\) which takes a point \(p \in S^{-}\)to the intersection of \(P\) with the line that connects \(p\) to the center of \(S^{2}\) is a geodesic mapping. c. Show that a surface of constant curvature admits a local geodesic mapping into the plane for every \(p \in S\).

Surfaces of Liouville are those surfaces for which it is possible to obtain a system of local coordinates \(\mathbf{x}(u, v)\) such that the coefficients of the first fundamental form are written in the form $$ E=G=U+V, \quad F=0 $$ where \(U=U(u)\) is a function of \(u\) alone and \(V=V(v)\) is a function of \(v\) alone. Observe that the surfaces of Liouville generalize the surfaces of revolution and prove that (cf. Example 5) a. The geodesics of a surface of Liouville may be obtained by integration in the form $$ \int \frac{d u}{\sqrt{U-c}}=\pm \int \frac{d v}{\sqrt{V+c}}+c_{1} $$ where \(c\) and \(c_{1}\) are constants that depend on the initial conditions. b. If \(\theta, 0 \leq \theta \leq \pi / 2\), is the angle which a geodesic makes with the curve \(v=\) const., then $$ U \sin ^{2} \theta-V \cos ^{2} \theta=\mathrm{const} $$ (Notice that this is the analogue of Clairaut's relation for the surfaces of Liouville.)

Let \(\mathbf{x}: U \subset R^{2} \rightarrow R^{3}\), where $$ \begin{aligned} U &=\left\\{(\theta, \varphi) \in R^{2} ; 0<\theta<\pi, 0<\varphi<2 \pi\right\\}, \\ \mathbf{x}(\theta, \varphi) &=(\sin \theta \cos \varphi, \sin \theta \sin \varphi, \cos \theta), \end{aligned} $$ be a parametrization of the unit sphere \(S^{2}\). Let $$ \log \tan \frac{1}{2} \theta=u, \quad \varphi=v, $$ and show that a new parametrization of the coordinate neighborhood \(\mathbf{x}(U)=V\) can be given by $$ \mathbf{y}(u, v)=(\operatorname{sech} u \cos v, \operatorname{sech} u \sin v, \tanh u) . $$ Prove that in the parametrization \(\mathbf{y}\) the coefficients of the first fundamental form are $$ E=G=\operatorname{sech}^{2} u, \quad F=0 $$ Thus, \(\mathbf{y}^{-1}: V \subset S^{2} \rightarrow R^{2}\) is a conformal map which takes the meridians and parallels of \(S^{2}\) into straight lines of the plane. This is called Mercator's projection.
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