Orientierbar

Orientierbar is a term used in geometry and topology to describe objects that admit an orientation. In this sense, an object is orientierbar if it is possible to choose a consistent notion of “handedness” throughout the object, so that the local orientations of its parts fit together globally.

In differential geometry, a differentiable manifold M is orientierbar if there exists a globally consistent orientation on all tangent spaces TM_p, varying continuously with p in M. Equivalently, there exists a nowhere vanishing top-degree differential form on M. Other equivalent formulations include the triviality of the orientation bundle and the vanishing of the first Stiefel-Whitney class w1(M). For vector bundles, a real vector bundle E is orientierbar if its structure group reduces to GL+(k, R); the tangent bundle of an orientierbarer manifold is orientierbar.

Examples illustrate the concept. The sphere S^n is orientierbar for every n. Real projective space RP^n is orientierbar iff n is odd. The Möbius band and the Klein bottle are non-orientierbar. The torus S^1 × S^1 is orientierbar.

Consequences of orientability include the ability to define a global volume form and to integrate differential forms without ambiguity. It also determines whether submanifolds or bundles admit globally consistent choices of orientation. Determining orientierbarkeit can be approached via the first Stiefel-Whitney class, or by seeking a global, continuous choice of orientation for tangent spaces; failure in these checks signals non-orientability.