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I have a question that I fear may raise some objection to the fact that it has been posted here, but I cannot think of a more appropriate place to pose it. I am not a mathematician; I'm a historian, and I am working on piece of prose writing that discusses the way in which an anthropologist from the early 20th century defined the idea of what a "culture" is. The idea here, of course, is that cultures can be defined differently, and in my work I want to argue that this anthropologist described cultures as "manifold" objects.

The question is to what extent is this description metaphorical and to what extent is it literal. In language, the definition of a manifold that I am interested in is the idea that a manifold is a whole thing with many distinct parts. I'm wondering, however, if the mathematical concept of a manifold may expand in important ways on this idea. What I'm wondering is what is a manifold object in mathematical terms.

My sense is that a manifold object in math is a way to talk about a space or an object that is continuous and yet which extends in dimensions that go beyond those that humans can with their basic senses (which are confined to euclidean space?) perceive?

I'm sure this definition falls harshly on many of your ears, but that is exactly why I am writing here. How can I understand/describe this concept in prose writing better? Am I completely off in my understanding? Any help would be wonderful!

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I think manifold just has two meanings, the mathematical meaning stands on its own, but the meaning in prose is very different. Do you think the anthropologist was using the topologists definition of the term? –  Thomas Andrews Sep 27 '13 at 12:36
    
In layman's terms, a manifold (in the mathematical sense, but Thomas is correct that there is another) is an object which, when looked at closely, appears to be a "flat" Euclidean space (think of looking at a sphere closely and percieving a flat plane), but may have interesting global structure (i.e., the sphere isn't actually a flat plane). The dimension is a bit of a red herring - in abstract mathematics (and indeed in a huge number of applications) there isn't anything special about dimensions smaller than 4. –  Matt Pressland Sep 27 '13 at 12:37
    
A manifold is a space (think of it as a shape maybe) which locally looks just like usual Euclidean space. That is, if you only look at the points which are a small distance away from any chosen point in the manifold, you wouldn't be able to tell the difference between a small volume in Euclidean space. It's how these small Euclidean 'neighbourhoods' are stitched which gives the rich structure to a manifold - the so called - 'global structure' of the manifold. –  Daniel Rust Sep 27 '13 at 12:38
    
Looking it up in google, I forgot the "machine" definition: "a pipe or chamber branching into several openings." I'd bet that's the meaning that caused mathematicians to use it for the topological spaces. –  Thomas Andrews Sep 27 '13 at 12:39
    
But I suspect the meaning the anthropologist intended was: "something with many different parts or forms, in particular." –  Thomas Andrews Sep 27 '13 at 12:41

5 Answers 5

A lot of interest in manifolds stems from the idea that they are "locally trivial" in a suitable sense. Really, the modern geometer's mantra is "locally trivial, globally complicated."

What does it mean to describe a "geometric object"? In high school, we all learn basic (Euclidean) geometry: the study of symmetries and transformations of solid objects in the plane ($\mathbb{R}^2$) or in space ($\mathbb{R}^3$). You'll see generalizations and abstractions of this in subjects like linear algebra or group theory. An advanced high-schooler or early undergraduate might encounter non-Euclidean geometries. The basic examples of this that still resemble geometry on the plane are geometry on the sphere and geometry on the saddle (i.e., hyperbolic paraboloid). To get an idea of why these are different, try drawing a "triangle" on these objects. What do you notice? Hint: if you add up the angles inside these triangles, do they equal $180^\circ$? What about "parallel lines"?

Really, a manifold, be it topological or smooth or complex or whatever, encapsulates the idea of a "nice" geometric object that, if you "zoom in" closely, is reminiscent of your basic high school geometry. All of high school math and maybe the first year or two of college level math deal with the geometry of Euclidean space, from many different perspectives. It is natural then to say, "okay, I get Euclidean space. What kind of objects will I encounter if I only require the space to be locally Euclidean? What can I salvage? What is lost?"

Cue 20th century math and beyond. You can do so much with just this small generalization. This is one of the big steps that was needed for Einstein's theory of general relativity: space is curved, not flat (at least, globally).

What you should take away from this: a manifold is a "sufficiently nice" space that is locally very easy to describe, but whose total or "global" structure might be all twisted up and complicated.

P.S. you might also be very interested in the notion of a "sheaf". Whenever there is a comparison between the local structure and global structure of a geometric object, the notion of sheaf is hiding in the background. And, shamelessly, I link you to a blog post of mine from a while back that describes these things from an intuitive perspective:

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Dear Brian, I am just now reading your description and finding it very helpful. Would you be able to point me to a publication of yours or someone else's that describes this -- in particular the connection to the idea of something being "locally trivial, globally complex? Something citable is what I mean. –  fraxture May 2 at 14:48

The easiest way to understand what a manifold is, is to consider the earth. When you look around you the earth looks flat, but when you zoom out, it is a sphere (a geoid, i guess, but who cares?).

More generally, a surface, or 2-manifold, is one that looks locally like a piece if $\mathbb{R}^2$. In other words, it is put together by "gluing" pieces of the plane together in a smooth manner. This is perhaps the origin of the term itself (I'm not sure).

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In prose: A manifold is a set with local coordinate information, built in such a way that if two local coordinate overlap somewhere, there is a "nice" transition function from the coordinates of the overlap in the first local coordinate system to the second system. For example, I can put local coordinates on a sphere by stereographic projections - you can look up pictures of this.

Metaphor: If I have multiple partial maps of a country, compiled by cartographers using instruments that are the same except for some predictable error, and I have a transition formula to go from one cartographers map to the map of another cartographer and back by only stretching or squeezing the maps each one provides, then I can use all of the information they provide me to understand something about the geometry of the country (depending on how "nice" the transition formula is), even those each piece was incomplete and at first seemingly incompatible with the others.

You can look up a rigorous definition here or on wikipedia. There is often a picture that comes with this definition of two overlapping disks on a surface, darkened where they overlap, arrows sending each disk onto two sections of a plane, and then another arrow sending the darkened bits in one planar section to the other. This is a useful diagram to assist in understanding this idea. You will see the word "homeomorphic" - roughly this just means a function that doesn't distort the shape too much, it doesn't cut it or glue bits together, it just squishes it around. You may see the word diffeomorphic - this just is just a function that doesn't distort the shape to much in a way that is more restrictive than a homeomorphism - all the squishing is continuously (or infinitely) differentiable, so you don't make any sharp corners, for instance.

PS I'm a student, not yet a mathematician, so take what I say with a grain of salt.

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The topological term, "manifold," is already handled above bout several answers, but I just want to say that the author almost certainly did not mean the topological term "manifold."

Without an exact quote, a "manifold object" could be using "manifold" as either an adjective or a noun. But the common non-mathematical meanings for "manifold" are, according to Google:

adjective: many and various

noun:
1. a pipe or chamber branching into several openings
2. (technical) something with many different parts or forms, in particular.

I suppose he could have been using any of these meanings, but I suspect either the adjective or the noun (2). He might have mean noun (1) in a metaphoric sense.

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With this comment in mind, I think what I may have been trying to understand is whether there is any overlap between the vernacular and mathematical definitions... –  fraxture May 2 at 14:53
    
The "pipe" term is probably the origin of the mathematical term - a pipe, idealized, is what mathematicians would call a 2-dimensional manifold, which were the manifolds that interested mathematicians first. @fraxture –  Thomas Andrews May 2 at 20:55

The original is always the best:

http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Geom/WKCGeom.html

Michael

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