# nLab meridian

Contents

### Context

#### Spheres

n-sphere

low dimensional n-spheres

# Contents

## Idea

Given an n-sphere $S^n$ for $n \,\geq 1\,$, regarded under its standard unit sphere embedding into Cartesian space

$S^n \;\subset\; \mathbb{R}^{n+1} \,\simeq\, \mathbb{R} \times \mathbb{R}^n \,,$

then a meridian is any geodesic (great half-circle) from the “north pole” $\big(1,\vec 0\big)$ to the “south pole” $\big(-1,\vec 0\big)$ (or more generally between any pair of antipodes).

Hence the $n$-sphere may be regarded as the union of

1. its north pole

2. its south pole

3. all meridians.

This decomposition exhibits the $n$-sphere as the (un-reduced) suspension of the $(n-1)$-sphere which is its equator.