4form

4form, or four-form, is a differential form of degree four on a differentiable manifold. It is a section of the fourth exterior power of the cotangent bundle, Λ^4(T^*M). A nonzero 4-form requires the manifold to have dimension at least four.

Locally, in a coordinate chart (x^1,...,x^n), any 4-form has the form α = ∑ f_{ijkl} dx^i ∧ dx^j ∧ dx^k ∧ dx^l, where i<j<k<l and the f_{ijkl} are smooth functions. The basic 4-forms dx^i ∧ dx^j ∧ dx^k ∧ dx^l form a basis for the space of 4-forms on a chart.

On an oriented 4-manifold, the 4-forms can be integrated over the manifold to obtain real numbers. Since d maps Λ^4 to Λ^5, and there are no 5-forms in four dimensions, every 4-form is closed (dα = 0). In higher dimensions, a 4-form need not be closed.

With a metric, the Hodge star operator identifies Λ^4 with Λ^0; in four dimensions any 4-form can be written as f vol, where vol is a chosen volume form and f is a smooth function. This decomposition is useful for pairing 4-forms with test functions or for expressing top-degree invariants.

Examples include the standard volume form on R^4, dx^1 ∧ dx^2 ∧ dx^3 ∧ dx^4. In geometry and physics, 4-forms occur as volume forms and as curvature-related densities (for example, Euler classes and certain topological terms) in four dimensions.

See also differential forms, exterior algebra, wedge product, Hodge theory, de Rham cohomology, four-manifold.