Orientierbare

Orientierbare, or orientable in English, describes a property of manifolds and related spaces: they admit a globally consistent choice of orientation for their tangent spaces.

Definition and equivalent formulations

For a differentiable manifold M of dimension n, orientierbar means there exists an orientation that varies continuously across M. Concretely, each tangent space TpM is given an orientation, and these local orientations can be chosen so that they fit together to form a global orientation. Equivalently, M carries a nowhere-vanishing top-degree differential form (a volume form). Another equivalent condition is that the transition maps between any two charts have positive determinant, i.e., are orientation-preserving. The choice of a global orientation can be viewed as selecting one of the two connected components of the oriented frame bundle of M. In cohomological terms, M is orientierbar if and only if its first Stiefel-Whitney class w1(M) vanishes in H^1(M; Z/2).

Examples and counterexamples

Spheres and tori are orientierbar: S^n and T^k admit global orientations. On the other hand, the Möbius strip, the Klein bottle and the real projective plane RP^2 are non-orientierbar. Among projective spaces, RP^n is orientierbar exactly when n is odd.

Boundary and orientation conventions

If M has a boundary, orientation is defined on its interior, and the boundary inherits an induced orientation by the outward normal first convention. Orientation is a fundamental invariant in geometry and topology, influencing integration, intersection theory, and the behavior of maps between manifolds.

German usage

In German mathematical texts, orientierbare Mannigfaltigkeiten are manifolds that are orientierbar; their orientation can be described in terms of volume forms, chart transitions, or Stiefel-Whitney classes.