Surface - Surfaces in Geometry

Surfaces in Geometry

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 diffeomorphic to some open set in E². 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 surfaces are classified up to diffeomorphism by their Euler characteristic and orientability.

Smooth surfaces equipped with Riemannian metrics are of fundational 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:

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.

Every closed orientable surface admits a complex structure. 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.

Read more about this topic:  Surface

Other articles related to "surfaces, surface":

Spray-and-vac Cleaning
... accomplished without the need for workers to touch soiled surfaces with their hands - used in professional cleaning in which a pressurized, diluted cleaning solution is applied to soiled or contaminated ... solids and dissolved contaminants that have been removed from the surface ... - 5,000 psi, spray-and-vac units are designed for long-term indoor use on grout and other surfaces, and offer variable application pressures ranging from 110 – 500 psi ...
Stray Light - Sources
... Reflections from lens surfaces ... radiation from the infrared detector reflected back to itself from lens surfaces ... Light scattered from the surfaces of supporting structures within the optical system ...
Siebel Si 202 - Design and Development
... variants had sharply clipped wing and tail surfaces, giving the Hummel an attractively angular appearance compared with its contemporaries ... a plywood covered wooden structure, as were the fixed tail surfaces, rudder and elevators being fabric covered ... The horizontal tail surfaces were set noticeably aft of the rudder, rather like the more recent Piper PA-28 ...

Famous quotes containing the words geometry and/or surfaces:

    ... geometry became a symbol for human relations, except that it was better, because in geometry things never go bad. If certain things occur, if certain lines meet, an angle is born. You cannot fail. It’s not going to fail; it is eternal. I found in rules of mathematics a peace and a trust that I could not place in human beings. This sublimation was total and remained total. Thus, I’m able to avoid or manipulate or process pain.
    Louise Bourgeois (b. 1911)

    But ice-crunching and loud gum-chewing, together with drumming on tables, and whistling the same tune seventy times in succession, because they indicate an indifference on the part of the perpetrator to the rest of the world in general, are not only registered on the delicate surfaces of the brain but eat little holes in it until it finally collapses or blows up.
    Robert Benchley (1889–1945)