nLab meridian

Redirected from "meridians".
Contents

Context

Spheres

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Manifolds and cobordisms

Contents

Idea

Given an n-sphere S nS^n for n1n \,\geq 1\,, regarded under its standard unit sphere embedding into Cartesian space

S n n+1× n, 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” (1,0)\big(1,\vec 0\big) to the “south pole” (1,0)\big(-1,\vec 0\big) (or more generally between any pair of antipodes).

Hence the nn-sphere may be regarded as the union of

  1. its north pole

  2. its south pole

  3. all meridians.

This decomposition exhibits the nn-sphere as the (un-reduced) suspension of the (n1)(n-1)-sphere which is its equator.

References

See also:

Last revised on January 5, 2023 at 19:12:58. See the history of this page for a list of all contributions to it.