reorientability

Reorientability is a property of a manifold, a mathematical space that locally resembles Euclidean space. A manifold is said to be reorientable if there is a consistent way to define an orientation across the entire manifold. An orientation essentially assigns a "handedness" to the tangent spaces at each point.

Consider a simple curve in the plane. We can choose to traverse it in a particular direction, say from left to right. This defines an orientation for the curve. For a surface, an orientation can be thought of as choosing which side is "up" or which direction the normal vector points. In higher dimensions, this concept generalizes to the determinant of a basis in the tangent space.

A manifold is reorientable if it's possible to make these orientation choices at every point such that they smoothly connect. If a manifold is not reorientable, any attempt to define a global orientation will inevitably lead to a contradiction. For instance, if you try to define an orientation on a Möbius strip, and you follow a path around its single edge, you will find that when you return to your starting point, the orientation you've carried with you has been flipped.

Familiar examples of reorientable manifolds include spheres, tori, and Euclidean spaces. The Klein bottle and the Möbius strip are classic examples of non-reorientable manifolds. The property of reorientability is crucial in many areas of mathematics and physics, particularly in differential geometry, topology, and fields like general relativity where orientation plays a significant role.