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A two dimensional space (eg. $\mathbb{R}^2$ ) could be a flat or it could be the surface of sphere (or of any shape) in a 3-dimensional space. How do we distinguish between such spaces without invoking higher dimensions.(here I am using the term 'space' in a general term and not strictly in a manner used in mathematics notations). What is the subject of math which deals with such properties ? Also please give the mathematical term closer to the term 'space'.

PS: please feel free to edit this post if it could make it more meaningful and clear.

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A metric is one of the standard ways, this produces things like intrinsic notions of curvature, such as Gauss or Sectional Curvature. –  Ryan Budney Dec 14 '10 at 3:58
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@Qiaochu Yuan : it is not very easily followable for me from the wikipedia. –  Rajesh D Dec 14 '10 at 4:08
    
@Rajesh: Riemannian structure is an extra structure one can put on manifolds to make sense of notions such as length, geodesics, and curvature. It is precisely the structure needed to distinguish flat R^2 from the R^2 obtained by removing a point from a sphere; the two are diffeomorphic as smooth manifolds but not isomorphic as Riemannian manifolds. The answer to the question "what is the subject of math which deals with such properties" is "Riemannian geometry." –  Qiaochu Yuan Dec 14 '10 at 4:13
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Rajesh, this is a very good question. The whole motivation behind the development of Riemannian geometry is precisely the desire to get rid of the necessity of embedding a geometric object into some Euclidean space before being able to talk about its geometric properties, like curvature. If you want to start reading about these things, start with an introductory book on differential geometry. –  Alex B. Dec 14 '10 at 4:16
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I recommend reading up on differential geometry.

I think what you are asking is how to determine if a space is curved without access to the embedding space. That depends on curvature of the space. There are some spaces that are equivalent intrinsically but different extrinsically (i.e. a cylinder or cone to a flat plane). In these spaces you cannot tell the difference from within the space.

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