states that the integral of the Gaussian curvature K over the entire surface S is determined by the Euler characteristic: This result exemplifies the deep relationship between the geometry and topology of surfaces (and, to a lesser extent, higher-dimensional manifolds). Another way in which surfaces arise in geometry is by passing into the complex domain. A complex one-manifold is a smooth oriented surface, also called a Riemann surface. Any complex nonsingular algebraic curve viewed as a complex manifold is a Riemann surface. In fact, every compact orientable surface is realizable as a Riemann surface. Thus compact Riemann surfaces are characterized topologically by their genus: 0, 1, 2, .... On the other hand, the genus does not characterize the complex structure. For example, there are uncountably many non-isomorphic compact Riemann surfaces of genus 1 (the elliptic curves). Complex structures on a closed oriented surface correspond to conformal equivalence classes of Riemannian metrics on the surface. One version of the uniformization theorem (due to Poincaré) states that any Riemannian metric on an oriented, closed surface is conformally equivalent to an essentially unique metric of constant curvature. This provides a starting point for one of the approaches to Teichmüller theory, which provides a finer classification of Riemann surfaces than the topological one by Euler characteristic alone. A complex surface is a complex two-manifold and thus a real four-manifold; it is not a surface in the sense of this article. Neither are algebraic curves defined over fields other than the complex numbers, nor are algebraic surfaces defined over fields other than the real numbers. See also Boundary (topology) Volume form, for volumes of surfaces in En Poincaré metric, for metric properties of Riemann surfaces Roman surface Boy's surface Tetrahemihexahedron Crumpled surface, a non-differentiable surface obtained by deforming (crumpling) a differentiable surface Notes References Simplicial proofs of classification up to homeomorphism , English translation of 1934 classic German textbook , Chapter I , Cambridge undergraduate course , for closed oriented Riemannian manifolds Morse theoretic proofs of classification up to diffeomorphism , careful proof aimed at undergraduates (Original 1969-70 Orsay course notes in French for "Topologie des Surfaces") Other proofs , similar to Morse theoretic proof using sliding of attached handles , short elementary proof using spanning graphs , contains short account of Thomassen's proof External links Classification of Compact Surfaces in Mathifold Project The Classification of Surfaces and the Jordan Curve Theorem in Home page