planeben

Planeben is a term used in geometry to denote a two-dimensional flat subspace embedded within a higher-dimensional ambient space. It is defined as a maximal, flat affine subspace of dimension two, in which the intrinsic curvature vanishes. Planebens are characterized by local isometry to the Euclidean plane and by the property that geodesics lying entirely in the subspace map to straight lines with respect to the inherited metric.

In Euclidean three-space, planebens coincide with ordinary planes. When the ambient space is curved, planebens appear as flat leaves within a foliation or as submanifolds with zero intrinsic curvature. They intersect along lines, and two distinct planebens that are not parallel meet, if at all, along a straight line contained in both.

Construction and use: Given a coordinate chart, the planeben through a point consists of all points obtained by flowing along straight lines in two fixed tangent directions. In practice, the concept provides a simplifying assumption for models that require flat geometry within a curved setting, and it serves as a reference surface for measurements and comparisons.

Applications: In speculative physics and computer graphics, planebens can model flat facets on curved surfaces, facilitate collision detection or texture mapping on curved manifolds, and aid in visualizing flat components of more complex geometries.

See also: planar geometry, submanifold, foliation, affine space.