# S^2 and homogeneous spaces

• Lonewolf
In summary, S^2 is a 2-dimensional sphere in 3-dimensional space with no edges or corners, often used in mathematical and scientific studies. A homogeneous space is a mathematical object with the same properties at every point, and S^2 is a prime example. They have various applications in mathematics, physics, and engineering and are studied using tools such as differential equations and group theory, and can be represented with mathematical models and visualizations.

#### Lonewolf

I've seen $S^2$ written as the quotient $SO(3)/SO(2)$. Can someone run me through how to show this, or point to somewhere that does, as I've only seen it stated?

It's called the Hopf fibration. There are many ways to realize it, one of which uses quarternions I recall. Anyway, a google for hopf fibration will give you more than I can remember off the top of my head, and there's even an animation illustrating it somewhere.

Thanks, it helped having something to search for!

## What is S^2?

S^2, also known as the 2-dimensional sphere, is a surface in 3-dimensional space that is defined by the equation x^2 + y^2 + z^2 = 1. It is a curved surface with no edges or corners, and is often used in mathematical and scientific studies.

## What is a homogeneous space?

A homogeneous space is a mathematical object that has the same properties at every point. This means that translations and rotations of the space will leave it unchanged. Examples of homogeneous spaces include spheres, planes, and tori.

## How are S^2 and homogeneous spaces related?

S^2 is a prime example of a homogeneous space. It is a curved surface with the same properties at every point, making it a perfect example of a homogeneous space. In fact, many properties and theorems that apply to homogeneous spaces can also be applied to S^2.

## What are some applications of S^2 and homogeneous spaces?

S^2 and homogeneous spaces have various applications in mathematics, physics, and engineering. They are used in the study of differential geometry, topology, and dynamical systems. They also have applications in computer graphics, robotics, and cosmology.

## How are S^2 and homogeneous spaces studied and represented?

S^2 and homogeneous spaces are typically studied using mathematical tools such as differential equations, group theory, and topology. They can also be represented using mathematical models and visualizations, such as graphs, diagrams, and computer simulations.