Manifolds 29 | Differential Forms

Differential forms on a smooth manifold are built by assembling alternating multilinear forms on each tangent space, then tracking how those local descriptions vary smoothly from point to point. Start with a smooth manifold M of dimension n. At every point P in M, the tangent space T_P M is an n-dimensional vector space, and an alternating k-form on that vector space is an object that eats k tangent vectors and flips sign appropriately under swaps.

The key construction is a field of such k-forms: a k-form on M assigns to each point P an alternating k-form on T_P M. In notation, this is written as Ω(P), meaning Ω at P lives in the space of alternating k-forms on T_P M. Once this pointwise assignment is in place, standard operations from multilinear algebra extend naturally. If Ω is a k-form and η is an s-form on M, their wedge product Ω ∧ η is defined pointwise: at each P, it is the usual wedge product of the alternating forms Ω(P) and η(P). The result is a (k+s)-form on M.

Pullbacks work similarly but require the differential of a map. Given a smooth map f: N → M between manifolds, a k-form Ω on M can be pulled back to a k-form f*Ω on N. The definition is pointwise: at each point in N, the alternating k-form is evaluated using the differential d f_P, which maps tangent vectors in T_P N to tangent vectors in T_{f(P)} M. This is the mechanism that replaces the “linear map” pullback used in pure vector-space settings.

To make these objects computable, the discussion then pins down local bases. Using a chart (with coordinates coming from a parameterization into R^n), one gets a coordinate basis {∂/∂x^1, …, ∂/∂x^n} for T_P M and a dual basis {dx^1, …, dx^n} for T_P^*M. The dual basis is characterized by the Kronecker delta pairing: dx^j(d x^k) = δ^j_k (one when indices match, zero otherwise). From the dual basis, a basis for alternating k-forms is formed by wedging k distinct dx’s with increasing indices. For example, on a 3-dimensional manifold, the basis of 2-forms consists of dx^1∧dx^2, dx^1∧dx^3, and dx^2∧dx^3—exactly the combinations with increasing indices.

With that basis, any k-form Ω can be written locally in a chart as a linear combination of wedge products of the basis 1-forms. The coefficients are scalar component functions Ω_{μ1…μk}(P), one for each increasing index tuple. This yields the final step toward “differential forms” in the smooth sense: Ω is called differentiable at P if, in some (equivalently any) chart around P, all its component functions are differentiable there. If Ω is differentiable at every point of M, it is a differential form (with the usual convention that smooth 0-forms are just smooth functions). The payoff is that these local coefficient functions provide the smooth structure needed for later integration on manifolds.