Manifold: what's the meaning of this name?

  • Thread starter Goldbeetle
  • Start date
210
1
Dear all,
I've always wondered where the name "manifold" comes from?
Any idea?
Thanks,
Goldbeetle
 
I think it come from the fact that one of the meanings is a 'thing with several possible shapes'. A bare manifold is essentially that, a thing you can add structure too or deform into several possible shapes. At least that's my understanding of this.
 

quasar987

Science Advisor
Homework Helper
Gold Member
4,771
7
Riemann, who was the first to talk of manifolds, called them (in german!) something like "multiply extended quantities"... probably having in mind that they would be objects who could locally be parametrized by many coordinates... a natural generalization of surfaces.
If "manifold" is not as good a translation of the word Riemann used for them as "multiply extended quantities" is, at least it has the merit of being brief!
 
607
0
So the English usage in mathematics is as a translation of the German "Mannigfaltigkeit"
 
The first time I saw it I mistook it for "mainfold",haha
 

tiny-tim

Science Advisor
Homework Helper
25,790
249
mannigfaltigkeit

Hi Goldbeetle! Hi g_edgar! :smile:
So the English usage in mathematics is as a translation of the German "Mannigfaltigkeit"
"mannigfaltig" seems to be the German for "diverse" "various" or "multifarious",

and "mannigfaltigkeit" for "diversity" "variety" or "manifoldness".

At http://en.wikipedia.org/wiki/User:Markus_Schmaus/Riemann" [Broken], Markus Schmaus says …
In (I) Riemann defines a "Mannigfaltigkeit" as consisting of the "Bestimmungsweisen" (ways of determination) of a "Größenbegriff" (concept of quantity), with "Bestimmungsweisen" being the points of a "stetige Mannigfaltigkeit" (continuous manifold). The "stetige Mannigfaltigkeit" is not described as being composed from smaller pieces, nor does he mention local flattness. In another, not translated, part he mentions colors and the locations of "Sinngegenstände" (objects of perception) as the only simple concepts giving rise to "stetige Mannigfaltigkeiten". At another point he calls the possible shapes of a spacial figure a "Mannigfaltigkeit".
… and illustrates this with German quotations (and his own English translations) from Riemann's http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Geom/" [Broken]
 
Last edited by a moderator:

HallsofIvy

Science Advisor
Homework Helper
41,711
876
I have always assumed that "manifold" was associated with "many dimensions".
 

Related Threads for: Manifold: what's the meaning of this name?

  • Posted
Replies
7
Views
3K
  • Posted
Replies
12
Views
5K
Replies
4
Views
6K
Replies
6
Views
4K
Replies
24
Views
3K
Replies
7
Views
3K

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving
Top