Manifolds 29 | Differential Forms [dark version]

Differential forms on a smooth manifold are built by assembling alternating multilinear forms on each tangent space, then tracking how those local pieces vary smoothly from point to point. Concretely, for a smooth manifold M of dimension n, each point P has an n-dimensional tangent space T_P M. An alternating k-form at P is an element of the vector space of alternating k-linear maps on T_P M, and the manifold-wide object is a rule Ω that assigns to every point P such an alternating k-form Ω(P) on T_P M.

Once this pointwise assignment is in place, the usual algebra of multilinear forms carries over. 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 equals Ω(P) ∧ η(P), producing a (k+s)-form. This keeps the familiar exterior-algebra structure while moving from a single vector space to the varying tangent spaces across the manifold.

Smooth maps between manifolds also induce operations on differential forms. Given a smooth function f: N → M and a k-form Ω on M, the pullback f*Ω becomes a k-form on N by using the differential d f_P at each point P ∈ N. The pullback is defined pointwise by composing Ω(f(P)) with d f_P, which is the correct replacement for the earlier pullback notion that only worked for linear maps.

To make these objects computable, the transcript builds the coordinate machinery. Starting from a chart (a parameterization) around each point, one gets a coordinate basis for the tangent space T_P M, typically denoted by ∂/∂x^1, ∂/∂x^2, … (written in the transcript as D_1, D_2, etc.). Dual to this is a basis of the cotangent space T_P^*M, denoted dx^1, dx^2, …, characterized by dx^j(d x^k)=δ^j_k (Kronecker delta). From the dual basis, a basis for k-forms is formed using wedge products of k distinct dx’s with increasing indices.

The key local description is that any k-form Ω on M can be written in a chart as a linear combination of these basis wedge products: Ω(P)=∑_{μ1<…<μk} Ω_{μ1…μk}(P) dx^{μ1}∧…∧dx^{μk}. The coefficients Ω_{μ1…μk}(P) are real-valued component functions on the chart domain, and their behavior determines smoothness.

A k-form is called differentiable at a point P if its component functions are differentiable at P, and it becomes a differential form when this holds at every point of M. The definition is consistent across charts: differentiability at P in one chart implies differentiability in any chart around P. The transcript also notes that 0-forms correspond to smooth functions, and the overall framework is kept in the C^∞ (smooth) setting, setting up concrete examples for the next installment.