# Was the term "manifold" (or its German equivalent) chosen with the verb "to fold" in mind?

Recently I came across several papers of Monge and Lagrange, around the end of the 18th century, considering developable surface as 'folded' planes, using specifically the word "plié" (i.e. folded). Now, as these surface are now considered as part of differential geometry, is there any connection between the term Manifold, coined by Riemann as "Mannigfaltigkeit" in 1854, and the terms fold "Falte" and folding "falten"?

By connection I mean not just a conceptual one but in fact something that is said explicitly (by Gauss, Riemann etc); i.e. that these mathematicians meant that the term "Mannigfaltigkeit" and the new domain of research (differential geometry) are inherently and deliberately connected to a folded space.

Many thanks!

Notion of manifold is often motivated today by examples of simple surfaces, including developable ones, so this is a natural guess, but it is unlikely to be the case historically. German Mannigfaltigkeit is associated not so much with geometric folding as with "many folded into one", alternatively translated as "multiplicity". Cantor even used Mannigfaltigkeit for general sets until 1890s, when he switched to Menge (literally "quantity"), although according to Husserl "this conception does not coincide with that of Riemann and as used elsewhere in the theory of geometry, according to which a manifold is a collection not of merely united, but also ordered elements, and on the other hand not merely united, but continuously connected elements".

Riemann himself in his lecture uses Mannigfaltigkeit as almost synonymous with "magnitude", saying that he set himself "the task of constructing the notion of a multiply extended magnitude", and invokes different intuitions when first using the word, Mannigfaltigkeit can equally be discrete:"according as there exists... a continuous path from one to another or not, they form a continuous or discrete manifoldness; the individual specialisations are called in the first case points, in the second case elements". He is also quite clear on his influences:"besides some very short hints on the matter given by Privy Councillor Gauss in his second memoir on Biquadratic Residues, in the Gottingen Gelehrte Anzeige, and in his Jubileebook, and some philosophical researches of Herbart, I could make use of no previous labours". Herbart is a German philosopher who clarified and popularized Kant's philosophy of space, he also influenced Grassman and his "science of magnitude".

The motivation he cites is related to complex analysis:"Such researches have become a necessity for many parts of mathematics, e.g., for the treatment of many-valued analytical functions; and the want of them is no doubt a chief cause why the celebrated theorem of Abel and the achievements of Lagrange, Pfaff, Jacobi for the general theory of differential equations, have so long remained unfruitful". Finally, the main focus of the lecture are local metric properties of manifolds, especially curvature, but developable surfaces are folded from flat ones, i.e. they are locally flat and have zero curvature. So they wouldn't be interesting for what Riemann has in mind.

The word "Mannigfaltigkeit" itself was not invented by Riemann. It is a word meaning (great) variety or richness of types, in an every day sense. It might be used to describe that there are a variety of different things of a related type, like many different trees, or something along these lines.

While I suppose there is an etymological connection, in the common usage of the word this idea seems rather lost. Thus, as long as there is no actual evidence for it, I would be skeptical at that connection.

A further piece of evidence for the lack of a relation to "fold" is the example of a "real world" occurrence of "continuous manifold" that Riemann gives in "Ueber die Hypothesen, welche der Geometrie zu Grunde liegen,", the article linked to in a comment, it is: colors.

For a general explication of Riemann was up to, see Conifold's nice answer.

I am not sure whether this question is about mathematics or etymology. In the latter case the answer is: Yes, the suffix -fold, -faltig, -fältig derives from the noun fold, Falte. You can look them up in any etymological dictionary, for example here: http://www.dwds.de/?qu=Falte

• thanks! I am aware that these suffixes derive from "fold". maybe i'll reformulate my question: do Riemann, Gauss etc were deliberately using the term "Mannigfaltigkeit" to indicate that the space might be folded (and in that case, are there references for that) or that they used it to symbolize another meaning? Commented Sep 4, 2015 at 10:41
• The paper in which Riemann introduced the concept of Mannigfaltigkeit is here: maths.tcd.ie/pub/HistMath/People/Riemann/Geom/Geom.html I have not had the patience to read it all, but perhaps others might wish to do so.
– fdb
Commented Sep 4, 2015 at 16:01

The roots of falten, Falte, Zwiefalt, Sorgfalt and Mannichfaltigkeit (Cantor's original use) or Mannigfaltigkeit (modern use) are the same but the question has to be answered by "definitely not". The original meaning of the verb "falten" (to fold) is no longer present in Zwiefalt = Zweifel (doubt) and Sorgfalt (care). Same holds for manifold = Mannigfaltigkeit = Vielfalt. The emphasize is on viel = many, multiple. Neither Riemann nor Cantor had a folded space in mind.