Deltaform

Deltaform is a term encountered in differential geometry and related areas to denote a class of distribution-valued differential forms that encode singular geometric data along submanifolds. They generalize ordinary differential forms by including delta-type singularities, allowing the representation of concentrated mass, interfaces, or intersection data within a smooth ambient space.

Locally, a delta-form is described as a Dirac delta distribution supported on a submanifold S of codimension k, multiplied by a smooth differential form on the ambient manifold. When paired with a smooth test form, the result yields the integral of the test form over S with the appropriate orientation and density given by the coefficient form. This interpretation aligns delta-forms with the language of currents, where the delta-form acts as the current of integration along S.

Construction and properties of delta-forms often involve modeling them as currents of integration along a submanifold or as wedge products with delta distributions defined by local defining functions for S. They are linear under addition and compatible with pullbacks and pushforwards under smooth maps that are transverse to the submanifold. Delta-forms are not smooth objects; they are treated within distribution theory and require careful handling when performing operations like wedge products or exterior differentiation.

Applications span geometric analysis and mathematical physics. They are used to encode localized sources, compute intersection numbers, and express localization phenomena in integrals. In physics, delta-forms can model defects or branes as delta-type sources in background fields, linking geometric and physical descriptions through duality concepts.

See also: Dirac delta distribution, currents, differential forms, Poincaré duality, geometric measure theory.