Wednesday, August 11, 2021

The notion of Orientability

Orientability is a property of surfaces in Euclidean space that measures whether it is possible to make a consistent choice of surface normal vector at every point. A choice of a normal vector allows one to use the right-hand rule to define a “clockwise” direction of loops in the surface, as needed by Stokes’ theorem for instance.

More generally, orientability of an abstract surface, or manifold, measures whether one can consistently choose a “clockwise” orientation for all loops in the manifold. Equivalently, a surface is orientable if a two-dimensional figure (such as ) in the space cannot be moved continuously on that surface and back to its starting point so that it looks like its own mirror image (

The notion of orientability can be generalised to higher-dimensional manifolds as well. A manifold is orientable if it has a consistent choice of orientation, and a connected orientable manifold has exactly two different possible orientations.

In this setting, various equivalent formulations of orientability can be given, depending on the desired application and level of generality. Formulations applicable to general topological manifold/ often employ methods of homology theory, whereas for differentiable manifolds more structure is present, allowing a formulation in terms of differential forms.

An important generalization of the notion of orientability of a space is that of orientability of a family of spaces parameterized by some other space (a fiber bundle) for which an orientation must be selected in each of the spaces which varies continuously with respect to changes in the parameter values.

Most surfaces we encounter in the physical world are orientable. Spheres, planes, and tori are orientable, for example. But Möbius strips, real projective planes, and Klein bottles are non-orientable. They, as visualized in 3-dimensions, all have just one side.

The real projective plane and Klein bottle cannot be embedded in R3, only immersed with nice intersections.

Note that locally an embedded surface always has two sides, so a near-sighted ant crawling on a one-sided surface would think there is an “other side”. The essence of one-sidedness is that the ant can crawl from one side of the surface to the “other” without going through the surface or flipping over an edge, but simply by crawling far enough.

A closely related notion uses the idea of covering space. For a connected manifold M take M∗, the set of pairs (x, o) where x is a point of M and o is an orientation at x; here we assume M is either smooth so we can choose an orientation on the tangent space at a point or we use singular homology to define orientation.

Then for every open, oriented subset of M we consider the corresponding set of pairs and define that to be an open set of M∗. This gives M∗ a topology and the projection sending (x, o) to x is then a 2-to-1 covering map. This covering space is called the orientable double cover, as it is orientable. M∗ is connected if and only if M is not orientable.

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