# What's manifold?

by KFC
Tags: manifold
 P: 369 Hi there, I find that the term 'manifold' appears in many book of statistical physics or classical mechanics while talking about phase space. I try to search the explanation on online but it is quite abstract and hard to understand what's manifold really refers to. Can anyway explain this a bit with a simplest picture? Thanks
 Sci Advisor P: 2,470 An n-dimensional manifold, in simplest terms, is a representation of non-Euclidean (in general) n-dimensional space that is locally Euclidean* at any point. In other words, you can define a local coordinate system anywhere, but not necessarily a global coordinate system. Keep in mind, I am oversimplifying a bit. For formal definition, look up pretty much any topology text.
 Sci Advisor P: 2,953 One can think of a Manifold as any set of objects which can be described by a set of coordinate charts (called an atlas). A coordinate chart is a mapping between your set of objects to some Euclidean space R^n (n is then the dimension of your manifold). There's really nothing much more to it than that. If you can describe your set of objects in such a way, then it's a manifold.
P: 8
What's manifold?

 Quote by K^2 An n-dimensional manifold, in simplest terms, is a representation of non-Euclidean (in general) n-dimensional space that is locally Euclidean* at any point. In other words, you can define a local coordinate system anywhere, but not necessarily a global coordinate system. Keep in mind, I am oversimplifying a bit. For formal definition, look up pretty much any topology text.
I am a bit confused. Isn't it the "definition" of differentiable manifold instead of manifold?
P: 2,470
 Quote by agostino981 I am a bit confused. Isn't it the "definition" of differentiable manifold instead of manifold?
Could be. Did I make an assumption that mapping is differentiable when I said that coordinate system could be defined? I think you might be right about that. I'm sure the OP would be dealing with such, but I probably shouldn't have defined it so narrowly anyways. Matterwave's explanation is a lot closer to a formal, general definition of a manifold.
 Math Emeritus Sci Advisor Thanks PF Gold P: 39,693 A "manifold" is a geometric object that is locally Euclidean. More precisely, an n dimensional manifold is a topological space, together with a set of "pairs", $\{(M_\alpha, \phi_\alpha)\}$, in which one member of each pair, $M_\alpha$, is an open set and the other member, $\phi_\alpha$, is a continuous function from $M_\alpha$ to Rn. We require, further, that if p is any point in the manifold there exist at least one $\alpha$ such that p is in $M_\alpha$ ( the open sets cover the manifold). We require, further, that in the intersection of two such sets, $M_\alpha$ and $M_\beta$, $\phi_\alpha o\phi_\beta$ be a homeomorphism from Rn to itself. In order to have a differentiable manifold, we require that, in the intersection of $M_\alpha$ and $M_\beta$, both $\phi_\alpha o \phi_\beta$ and $\phi_\beta o\phi_\alpha$ be differentiable.
 Sci Advisor P: 2,470 Halls, OP states that he looked at former definitions. I think he was looking for a simpler illustration.
Homework
 Quote by HallsofIvy We require, further, that in the intersection of two such sets, $M_\alpha$ and $M_\beta$, $\phi_\alpha o\phi_\beta$ be a homeomorphism from Rn to itself.