doubleforms

Double forms are a way to organize antisymmetric tensors with two separate degrees. Let V be a finite-dimensional vector space over R or C. A double form of degrees (p, q) is an element of Λ^p V* ⊗ Λ^q V*. The collection of all double forms is the direct sum ⊕_{p,q} Λ^p V* ⊗ Λ^q V*. Conceptually, a (p, q)-double form acts as a multilinear map that is alternating in the first p arguments and alternating in the last q arguments, i.e., it is a bilinear form on Λ^p V × Λ^q V.

Operations on double forms include the exterior product: for α ∈ Λ^p V*, β ∈ Λ^q V*, α' ∈ Λ^{p'} V*, β' ∈ Λ^{q'} V*, we have (α ⊗ β) ∧ (α' ⊗ β') = (α ∧ α') ⊗ (β ∧ β'). This extends linearly to all double forms and increases the degrees additively.

With a choice of inner product (metric) on V, one can define contraction operations that reduce the degrees: contracting one index in each factor yields a map Λ^p V* ⊗ Λ^q V* → Λ^{p-1} V* ⊗ Λ^{q-1} V*. Repeated contractions yield traces or scalars.

In differential geometry, double forms provide a natural language for curvature-type objects. The Riemann curvature tensor can be viewed as a (2,2)-double form R ∈ Λ^2 V* ⊗ Λ^2 V* with its standard symmetries. The Ricci tensor and scalar curvature arise from appropriate contractions of R. The formalism also underpins constructions like the Kulkarni–Nomizu product and facilitates algebraic manipulations of curvature operators.

Applications of double forms appear in Riemannian geometry, global analysis, and the study of curvature invariants, offering a compact framework for handling multilinear, alternating structures.