Orientable

Orientable is a property in geometry and topology that applies to surfaces and, more generally, to manifolds. A surface is orientable if it is possible to make a consistent choice of orientation at every point; in practical terms, one can assign a "handedness" (such as clockwise versus counterclockwise) that varies continuously across the surface without ever flipping when moving along it. Equivalently, an orientable surface admits a nowhere-vanishing continuous normal vector field, or, in the language of calculus, a nowhere-vanishing top-dimensional differential form. Another common criterion is that the surface is two-sided: a loop in the surface cannot force you to switch to the opposite side of the surface when you parallel-transport a normal vector around the loop.

In two dimensions, orientability is often described by the existence of a globally consistent sense of rotation. If such a global choice exists, the surface is orientable; if not, it is non-orientable. A non-orientable surface cannot be consistently assigned a global "up" direction.

Common examples: the sphere, the plane, and the torus are orientable. Non-orientable examples include the Möbius strip, the real projective plane, and the Klein bottle. The Möbius strip, in particular, is the familiar one-sided surface that has only one boundary component and no consistent two-sided orientation.

Orientability plays a fundamental role in geometry and topology, influencing how surfaces can be integrated over, how they admit maps that preserve orientation, and how their global structure is classified. In higher dimensions, orientability generalizes in parallel and is closely related to the existence of global orientation forms and to properties of vector bundles.