The product geometry $S^2 \times \mathbb E$

What is $\mathbb S^2\times\mathbb{E} $

This geometry is the cartesian product of the two-sphere and the real line. It can be also seen as the universal cover of the product $M = S^2 \times S^1$ where $S^2$ is the two-sphere and $S^1$ the unit circle.

Click on the button below to reveal a concrete model of $S^2 \times \mathbb E$.

The two-sphere is the unit sphere in the 3-dimensional euclidean space. It can be described as $$ Y = \left\{ (x,y,z) \in \mathbb R^3 \mid x^2 + y^2 + z^2 = 1 \right\}$$ endowed with the following rieamanian metric $$ ds^2 = dx^2 + dy^2 + dz^2.$$

Consequently a possible model of $S^2 \times \mathbb E$ is the following subset $X$ of $\mathbb R^4$ $$ X = \left\{ (x,y,z,w) \in \mathbb R^3 \mid x^2 + y^2 + z^2 = 1 \right\}$$ endowed with the following rieamanian metric $$ ds^2 = dx^2 + dy^2 + dz^2 + dw^2.$$ The isometry group of $X$ is $O(3) \times {\rm Isom}(\mathbb R)$ where $O(3)$ is the isometry group of $S^2$ and ${\rm Isom}(\mathbb R) = \mathbb R \rtimes \mathbb Z/ 2 \mathbb Z$ is the isometry group of the real line.

Some views of $\mathbb S^2\times\mathbb{E} $

Warning: Some of the real-time simulations below requires a powerful graphic card. If your computer is not fast enough, you can reduce the size of your browser window. Click on the button below to reveal the fly commands.

The default controls to fly in the scene are the following. If you have a different keyboard, the keys should be the ones having the same location as the given ones on a QWERTY keyboard.

Command	QWERTY keyboard	AZERTY keyboard
Yaw left	a	q
Yaw right	d	d
Pitch up	w	z
Pitch down	s	s
Roll left	q	a
Roll right	e	e
Move forward	arrow up	arrow up
Move backward	arrow down	arrow down
Move to the left	arrow left	arrow left
Move the the right	arrow right	arrow right
Move upwards	'	ù
Move downwards	/	=