Surfaces in geometry Surface (topology)

polyhedra, such boundary of cube, among first surfaces encountered in geometry. possible define smooth surfaces, in each point has neighborhood diffeomorphic open set in e. elaboration allows calculus applied surfaces prove many results.

two smooth surfaces diffeomorphic if , if homeomorphic. (the analogous result not hold higher-dimensional manifolds.) closed surfaces classified diffeomorphism euler characteristic , orientability.

smooth surfaces equipped riemannian metrics of foundational importance in differential geometry. riemannian metric endows surface notions of geodesic, distance, angle, , area. gives rise gaussian curvature, describes how curved or bent surface @ each point. curvature rigid, geometric property, in not preserved general diffeomorphisms of surface. however, famous gauss–bonnet theorem closed surfaces states integral of gaussian curvature k on entire surface s determined euler characteristic:

∫

s


k

d
a
=
2
π
χ
(
s
)
.


{\displaystyle \int _{s}k\;da=2\pi \chi (s).}

this result exemplifies deep relationship between geometry , topology of surfaces (and, lesser extent, higher-dimensional manifolds).

another way in surfaces arise in geometry passing complex domain. complex one-manifold smooth oriented surface, called riemann surface. complex nonsingular algebraic curve viewed complex manifold riemann surface.

every closed orientable surface admits complex structure. complex structures on closed oriented surface correspond conformal equivalence classes of riemannian metrics on surface. 1 version of uniformization theorem (due poincaré) states riemannian metric on oriented, closed surface conformally equivalent unique metric of constant curvature. provides starting point 1 of approaches teichmüller theory, provides finer classification of riemann surfaces topological 1 euler characteristic alone.

a complex surface complex two-manifold , real four-manifold; not surface in sense of article. neither algebraic curves defined on fields other complex numbers, nor algebraic surfaces defined on fields other real numbers.