Editing Surface (topology) (section)

==Surfaces in geometry==
{{main|Differential geometry of surfaces}}

[[Polyhedron|Polyhedra]], such as the boundary of a [[cube]], are among the first surfaces encountered in geometry. It is also possible to define ''smooth surfaces'', in which each point has a neighborhood [[diffeomorphism|diffeomorphic]] to some open set in '''E'''<sup>2</sup>. This elaboration allows [[calculus]] to be applied to surfaces to prove many results.

Two smooth surfaces are diffeomorphic if and only if they are homeomorphic. (The analogous result does not hold for higher-dimensional manifolds.) Thus [[#Closed surface|closed surface]]s are classified up to diffeomorphism by their Euler characteristic and orientability.

Smooth surfaces equipped with [[Riemannian metric]]s are of foundational importance in [[differential geometry]]. A Riemannian metric endows a surface with notions of [[geodesic]], [[distance]], [[angle]], and area. It also gives rise to [[Gaussian curvature]], which describes how curved or bent the surface is at each point. Curvature is a rigid, geometric property, in that it is not preserved by general diffeomorphisms of the surface. However, the famous [[Gauss–Bonnet theorem]] for closed surfaces states that the integral of the Gaussian curvature ''K'' over the entire surface ''S'' is determined by the Euler characteristic:
:<math>\int_S K \; dA = 2 \pi \chi(S).</math>
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 curve#Elliptic curves over the complex numbers|elliptic curves]]).

Complex structures on a closed oriented surface correspond to [[conformally equivalent|conformal equivalence classes]] of Riemannian metrics on the surface. One version of the [[uniformization theorem]] (due to [[Henri Poincaré|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 [[field (mathematics)|field]]s other than the complex numbers,
nor are algebraic surfaces  defined over [[field (mathematics)|field]]s other than the real numbers.