 It was not immediately appreciated that Gauss’s approach allowed mathematicians to define surfaces as regions of the plane with a particular metric that are not to be obtained from surfaces in Euclidean 3-dimensional space. If one defines a surface as the image of a map from a piece of \(\mathbb{R}^2\) to \(\mathbb{R}^3\), then of course it is in \(\mathbb{R}^3\). But if one defines a surface as a region of \(\mathbb{R}^2\) with a particular metric, then there may be no surface in \(\mathbb{R}^3\) to which it corresponds. The first person to appreciate this seems to have been Riemann, who also extended this idea to any number of dimensions.