Editing Möbius strip (section)

===Surfaces of constant curvature===
The open Möbius strip is the [[relative interior]] of a standard Möbius strip, formed by omitting the points on its boundary edge. It may be given a [[Riemannian geometry]] of constant positive, negative, or zero [[Gaussian curvature]]. The cases of negative and zero curvature form geodesically complete surfaces, which means that all [[geodesic]]s ("straight lines" on the surface) may be extended indefinitely in either direction.

;Zero curvature
:An open strip with zero curvature may be constructed by gluing the opposite sides of a plane strip between two parallel lines, described above as the tautological line {{nowrap|bundle.{{r|dundas}}}} The resulting metric makes the open Möbius strip into a (geodesically) complete flat surface (i.e., having zero Gaussian curvature everywhere). This is the unique metric on the Möbius strip, up to uniform scaling, that is both flat and complete. It is the [[Quotient space (topology)|quotient space]] of a plane by a [[glide reflection]], and (together with the plane, [[cylinder]], [[torus]], and [[Klein bottle]]) is one of only five two-dimensional complete {{nowrap|[[flat manifold]]s.{{r|godinho-natario}}}}
;Negative curvature
:The open Möbius strip also admits complete metrics of constant negative curvature. One way to see this is to begin with the [[Poincaré half-plane model|upper half plane (Poincaré) model]] of the [[Hyperbolic geometry|hyperbolic plane]], a geometry of constant curvature whose lines are represented in the model by semicircles that meet the <math>x</math>-axis at right angles. Take the subset of the upper half-plane between any two nested semicircles, and identify the outer semicircle with the left-right reversal of the inner semicircle. The result is topologically a complete and non-compact Möbius strip with constant negative curvature. It is a "nonstandard" complete hyperbolic surface in the sense that it contains a complete hyperbolic [[half-plane]] (actually two, on opposite sides of the axis of glide-reflection), and is one of only 13 nonstandard {{nowrap|surfaces.{{r|cantwell-conlon}}}} Again, this can be understood as the quotient of the hyperbolic plane by a glide {{nowrap|reflection.{{r|stillwell}}}}
;Positive curvature
:A Möbius strip of constant positive curvature cannot be complete, since it is known that the only complete surfaces of constant positive curvature are the sphere and the [[Real projective plane|projective plane]].{{r|godinho-natario}} However, in a sense it is only one point away from being a complete surface, as the open Möbius strip is homeomorphic to the [[once-punctured]] projective plane, the surface obtained by removing any one point from the projective {{nowrap|plane.{{r|seifert-threlfall}}}}

The [[minimal surface]]s are described as having constant zero [[mean curvature]] instead of constant Gaussian curvature. The Sudanese Möbius strip was constructed as a minimal surface bounded by a great circle in a 3-sphere, but there is also a unique complete (boundaryless) minimal surface immersed in Euclidean space that has the topology of an open Möbius strip. It is called the Meeks Möbius {{nowrap|strip,{{r|lopez-martin}}}} after its 1982 description by [[William Hamilton Meeks, III]].{{r|meeks}} Although globally unstable as a minimal surface, small patches of it, bounded by non-contractible curves within the surface, can form stable embedded Möbius strips as minimal {{nowrap|surfaces.{{r|systolic}}}} Both the Meeks Möbius strip, and every higher-dimensional minimal surface with the topology of the Möbius strip, can be constructed using solutions to the [[Björling problem]], which defines a minimal surface uniquely from its boundary curve and tangent planes along this {{nowrap|curve.{{r|bjorling}}}}