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 Again, this can be understood as the quotient of the hyperbolic plane by a glide 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 projective plane. 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 The minimal surfaces 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 after its 1982 description by William Hamilton Meeks, III. 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 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 Spaces of lines The family of lines in the plane can be given the structure of a smooth space, with each line represented as a point in this space. The resulting space of lines is topologically equivalent to the open Möbius One way to see this is to extend the Euclidean plane to the real projective plane by adding one more line, the line at infinity. By projective duality the space of lines in the projective plane is equivalent to its space of points, the projective plane itself. Removing the line at infinity, to produce the space of Euclidean lines, punctures this space of projective Therefore, the space of Euclidean lines is a punctured projective plane, which is one of the forms of the