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Orientability

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This article discusses orientability and orientation on surfaces and manifolds. For orientation of vector spaces see orientation (mathematics). For alternate uses, see orientation.

In geometry and topology, a surface in <math>\mathbb{R}^3<math> is called non-orientable, if a figure such as the letter "R" can be moved about on the surface so that it becomes mirror-reversed. Otherwise the surface is said to be orientable.

Examples in low dimensions

Surfaces we normally encounter in every day life are orientable. For example, sphere, plane, torus. Example of non-orientable surfaces are Möbius strip, real projective plane, Klein bottle. These surfaces as visualized in 3-dimensions all have just one-side. 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. (Caveat: the real projective plane and Klein bottle can't be embedded in <math>\mathbb R^3<math>, only immersed with nice intersections.)

In general, the property of being orientable is not equivalent to being two-sided; however, this holds when the ambient space (such as <math>\mathbb{R}^3<math> above) is orientable. For example, a torus embedded in <math>K^2 \times S^1<math> can be one-sided, and a Klein bottle in the same space can be two-sided; here <math>K^2<math> refers to the Klein bottle.

Orientation by a triangulation

Orientability, for surfaces, is easily defined, regardless of whether the surface is embedded in an ambient space or not. Any surface has a triangulation: a decomposition into triangles such that each edge on a triangle is glued to at most one other edge. We can orient each triangle, by choosing a direction for each edge (think of this as drawing an arrow on each edge) so that the arrows go from head to tail as we go around the boundary of the triangle. If we can do this so that in addition triangles sharing an edge have arrows on that edge going in opposite directions, then we call what we've done an orientation for the surface. Note that whether the surface is orientable is independent of triangulation; this fact is not obvious, but a standard exercise.

This rather precise definition is based on intuition gathered from observing the following phenomenon:

Imagine a capital "R" written on the surface, that can freely slide along the surface but cannot be lifted off the surface (that letter is chosen because of its asymmetry). If the surface is a Möbius band, and the letter slides all the way around the band and returns to its starting point, then it will look like a mirror-image of an "R" rather than the "R" it looked like originally. If the surface is a sphere, on the other hand, that cannot happen.

The relation to the definition above is that sliding the "R" around from triangle to triangle in a triangulation gives an orientation for each triangle; the "R" in a triangle induces an obvious choice of arrow for each edge. The only obstruction to consistently orienting all the triangles is that when the "R" returns to its original starting triangle, it may induce choices of arrows going opposite to the original choice. Clearly, if this never happens, then we want the surface to be orientable, whereas if this does happen, then we want to call the surface non-orientable.

The definition above can be generalized to an n-manifold that has a triangulation, but there are problems with that approach: some 4-manifolds do not have a triangulation, and in general for n > 4 some n-manifolds have triangulations that are inequivalent.

Orientation by top-dimensional forms

Another way of thinking about orientability is thinking of it as a choice of "right handedness" vs. "left handedness" at each point in the manifold.

Formally, a <math>n<math>-dimensional differentiable manifold is called orientable if it possesses a differential form <math>\omega<math> of degree <math>n<math> which is nonzero at every point on the manifold. Conversely, given such a form <math>\omega<math>, we say that the manifold is oriented by <math>\omega<math>.

The crucial point to observe here is that such a differential form gives a choice of "right handed" basis at each point. A traveler in an orientable manifold will never change his/her handedness by going on a round trip.

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