Horizontality

Horizontality is the quality or condition of being horizontal: oriented parallel to the local horizon and perpendicular to the direction of gravity. In everyday use it describes lines, planes, or layouts that are level or aligned with a reference horizon.

In mathematics, horizontality is a formal concept in differential geometry and the theory of fiber bundles. Given a connection on a fiber bundle, the tangent space at each point splits into a vertical subspace (tangent to the fiber) and a horizontal subspace determined by the connection. Vectors in the horizontal subspace are called horizontal, representing directions that can be transported along the base without leaving the prescribed connection. A differential form is horizontal if it vanishes on vertical vectors. This framework underpins the construction of horizontal lifts and the study of parallel transport.

In physics and related mathematical formalisms, the term horizontality condition denotes a constraint that ties curvature to directions associated with an extended or auxiliary structure, such as a superspace in BRST or gauge-theory formalisms. The condition often serves to encode gauge invariance and relationships between physical and ghost fields in a compact, symmetry-driven way.

Horizontality also appears in applied fields such as surveying, architecture, and computer graphics, where ensuring horizontal alignment helps produce level surfaces, stable designs, or realistic renderings. Across disciplines, the precise meaning of horizontality depends on context, ranging from a geometric decomposition of space to a formal constraint in theoretical models.