Orientationpreserving

Orientation-preserving is a term used in geometry and topology to describe a transformation or map that maintains the chosen orientation of a space.

In Euclidean space, a differentiable map f: U -> R^n is orientation-preserving at a point if its Jacobian determinant det(Df(x)) is positive there; globally, f is orientation-preserving on U if det(Df(x)) > 0 for all x in U. For maps between oriented manifolds, orientation preservation means the differential sends every oriented basis of the tangent space to another oriented basis, i.e., it preserves the orientation class.

Properties include that the composition of orientation-preserving maps is orientation-preserving, and the inverse of an orientation-preserving bijection is orientation-preserving. In the context of manifolds, an orientation-preserving diffeomorphism preserves the orientation of the manifold, while an orientation-reversing one does the opposite.

Examples include rotations in the plane, which preserve orientation, whereas reflections reverse it. On the circle, a homeomorphism is orientation-preserving if it preserves the cyclic order. In one dimension, a diffeomorphism of the real line is orientation-preserving if it is increasing.

Applications of orientation preservation appear across differential topology, dynamical systems, and degree theory, where the sign of the Jacobian or the effect on orientation classes influences invariants such as degree and fixed-point phenomena.