Manifold: what's the meaning of this name?

Goldbeetle

Dear all,
I've always wondered where the name "manifold" comes from?
Any idea?
Thanks,
Goldbeetle

FredericGos

Let me google it for you... ;)

http://www.thefreedictionary.com/manifold

Goldbeetle

Quote by FredericGos

Let me google it for you... ;)

http://www.thefreedictionary.com/manifold

Thanks, maybe my question was not clear. The question is why was that word, "manifold", with those meanings (see your link) used to label this topological space

FredericGos

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

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!

g_edgar

So the English usage in mathematics is as a translation of the German "Mannigfaltigkeit"

einsteinyjn

The first time I saw it I mistook it for "mainfold",haha

tiny-tim

Hi Goldbeetle! Hi g_edgar!

Quote by g_edgar

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:Ma...chmaus/Riemann, 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 On the Hypotheses which lie at the Bases of Geometry

HallsofIvy

I have always assumed that "manifold" was associated with "many dimensions".

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Goldbeetle	Manifold: what's the meaning of this name? Dear all, I've always wondered where the name "manifold" comes from? Any idea? Thanks, Goldbeetle

FredericGos	Let me google it for you... ;) http://www.thefreedictionary.com/manifold

g_edgar	So the English usage in mathematics is as a translation of the German "Mannigfaltigkeit"

einsteinyjn	The first time I saw it I mistook it for "mainfold",haha

HallsofIvy Recognitions: Gold Member Science Advisor Staff Emeritus	I have always assumed that "manifold" was associated with "many dimensions".