nonflat

Nonflat is an adjective used to describe spaces, surfaces, or spacetimes that are not flat. In mathematics and physics, flatness typically refers to zero intrinsic curvature, meaning the space is locally indistinguishable from ordinary Euclidean space. A nonflat space, by contrast, has curvature that is nonzero at least at some location.

In differential geometry, flat spaces are those whose Riemann curvature tensor vanishes everywhere. Nonflat spaces include a wide range of geometries, such as spheres with positive curvature and hyperbolic spaces with negative curvature. The curvature determines how parallel lines behave, how triangles sum their angles, and how areas grow with radius. These properties contrast with flat Euclidean space, where parallel lines stay parallel, triangle angle sums equal 180 degrees, and area relations follow the familiar formulas.

In physics, the term often appears in general relativity and cosmology. Nonflat spacetime indicates the presence of gravity, which curves spacetime around mass and energy. Spatial sections of the universe can be nonflat, corresponding to closed (positive curvature) or open (negative curvature) geometries, though current measurements favor a universe very close to flat.

Applications of nonflat geometry appear in computer graphics, architecture, and the study of manifolds in pure mathematics. Understanding nonflat spaces involves tools from Riemannian geometry, such as curvature tensors and geodesics, and it yields insights into how space is shaped and how objects move within it.

See also: curvature, Gaussian curvature, Riemannian geometry, general relativity, cosmology.