Klein surface

In mathematics, a Klein surface is a <a href="/facts/Dianalytic_manifold/Kmm5qn9F">dianalytic manifold</a> of complex dimension 1. Klein surfaces may have a <a href="/facts/Manifold/rXPERJtu">boundary</a> and need not be <a href="/facts/Orientable_manifold/1SEQM2Pt">orientable</a>. Klein surfaces generalize <a href="/facts/Riemann_surfaces/qoj8ogv8">Riemann surfaces</a>. While the latter are used to study algebraic curves over the <a href="/facts/Complex_number/2w7mqI7e">complex numbers</a> analytically, the former are used to study algebraic curves over the real numbers analytically. Klein surfaces were introduced by <a href="/facts/Felix_Klein/WmL9guqK">Felix Klein</a> in 1882.
A Klein surface is a <a href="/facts/Surface_(topology)/jK1b1KIG">surface</a> (i.e., a <a href="/facts/Differentiable_manifold/aSnbXKcV">differentiable manifold</a> of real dimension 2) on which the notion of angle between two <a href="/facts/Tangent_vector/UEq5yffm">tangent vectors</a> at a given point is well-defined, and so is the angle between two intersecting curves on the surface. These angles are in the range [0,π]; since the surface carries no notion of orientation, it is not possible to distinguish between the angles α and −α. (By contrast, on Riemann surfaces are oriented and angles in the range of (-π,π] can be meaningfully defined.) The length of curves, the area of submanifolds and the notion of <a href="/facts/Geodesic/fbWR6l80">geodesic</a> are not defined on Klein surfaces.
Two Klein surfaces X and Y are considered equivalent if there are conformal (i.e. angle-preserving but not necessarily orientation-preserving) differentiable maps f:X→Y and g:Y→X that map boundary to boundary and satisfy fg = idY and gf = idX.

Klein surface open-in-new

Klein surface