Flat (geometry)

In geometry, a flat is an affine subspace, i.e. a subset of an affine space that is itself an affine space.^[1] Particularly, in the case the parent space is Euclidean, a flat is a Euclidean subspace which inherits the notion of distance from its parent space.

In an n-dimensional space, there are k-flats of every dimension k from 0 to n; flats one dimension lower than the parent space, (n − 1)-flats, are called hyperplanes.

The flats in a plane (two-dimensional space) are points, lines, and the plane itself; the flats in three-dimensional space are points, lines, planes, and the space itself. The definition of flat excludes non-straight curves and non-planar surfaces, which are subspaces having different notions of distance: arc length and geodesic length, respectively.

Flats occur in linear algebra, as geometric realizations of solution sets of systems of linear equations.

A flat is a manifold and an algebraic variety, and is sometimes called a linear manifold or linear variety to distinguish it from other manifolds or varieties.