Parallelform

Parallelform, often written as parallel form, is a differential form that remains parallel with respect to a connection on a differentiable manifold. A k-form ω is called parallel if its covariant derivative ∇ω vanishes identically.

Equivalently, ω is parallel if its components in any frame that is parallel along curves are constant. The space of parallel k-forms on a manifold M with connection ∇ is isomorphic to the subspace of Λ^k T^*_p M fixed by the holonomy group of ∇; thus the dimension of the space of parallel forms is dictated by the holonomy representation.

On Euclidean space with the standard flat connection, every constant coefficient k-form is parallel. If the manifold has nontrivial holonomy, parallel forms are more restricted. Manifolds with special holonomy, such as Calabi–Yau, G2, and Spin(7) manifolds, often admit nonzero parallel forms that define calibrated geometries and constrain the manifold’s structure.

In the Levi-Civita setting of a Riemannian manifold, a parallel form is automatically closed and coclosed, reflecting compatibility with the metric. Parallel forms therefore provide global invariants tied to the holonomy group and can influence topological and geometric properties.

Applications include the study of holonomy, calibrations, and the construction of geometric structures with restricted holonomy. They also play a role in global analysis, differential topology, and the theory of special geometric structures.

See also: differential form, covariant derivative, holonomy, calibration, special holonomy.