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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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    http://en.wikipedia.org/wiki/Theorema_Egregium2010-12-14

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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.