Tensor Fields and Differential Forms

Tensors are the multilinear objects of geometry; differential forms are their antisymmetric part, the natural integrands on a manifold. This page assembles the tensor and form calculus — the exterior derivative $d$, pullback, interior product — that every later page uses, building on the exterior-algebra basics of analysis/differential-forms.md and the vector fields of vector-fields-flows.md. The topological payoff is de-rham.md.

Tensor fields

A tensor of type $(r,s)$ at $p$ is a multilinear map $$T:\underset{r}{\underbrace{T_p^{\ast }M\times \cdots }}\times \underset{s}{\underbrace{T_{p}M\times \cdots }}\to \mathbb{R},$$ an element of $T_p^{(r,s)}M=(T_{p}M{)}^{\otimes r}\otimes (T_p^{\ast }M{)}^{\otimes s}$. A tensor field is a smooth section of the corresponding tensor bundle; in coordinates $$T=T^{i_1\cdots i_r}{}_{j_1\cdots j_s}{\partial }_{i_1}\otimes \cdots \otimes d{x}^{j_1}\otimes \cdots .$$ Vectors are $(1,0)$, covectors/1-forms are $(0,1)$, the metric is a symmetric $(0,2)$ tensor. Tensors transform by the appropriate Jacobian factors under a coordinate change — the property that makes tensor equations coordinate-independent.

Differential forms

A differential $p$-form is a totally antisymmetric $(0,p)$ tensor field — a smooth section of ${\Lambda }^p{T}^{\ast }M$. Locally $$\omega =\frac{1}{p!}{\omega }_{i_1\cdots i_p}d{x}^{i_1}\wedge \cdots \wedge d{x}^{i_p},$$ with ${\omega }_{\cdots }$ antisymmetric. The wedge product $\wedge$ makes ${\Omega }^{\bullet }(M)={⨁}_p{\Omega }^p(M)$ a graded-commutative algebra, $\alpha \wedge \beta =(-1{)}^{pq}\beta \wedge \alpha$ (see analysis/differential-forms.md). On an $n$-manifold, $p$ ranges $0$ (functions) to $n$ (top forms); ${\Omega }^p=0$ for $p>n$.

Three operations

Three natural operations act on forms:

• Exterior derivative $d:{\Omega }^p\to {\Omega }^{p+1}$: the unique antiderivation with $df={\partial }_{i}fd{x}^i$ on functions, satisfying $$d(\alpha \wedge \beta )=d\alpha \wedge \beta +(-1{)}^p\alpha \wedge d\beta ,\boxed{d^2=0}.$$ It unifies grad, curl, div (see analysis/vector-calculus.md); $d^2=0$ encodes "curl grad $=0$" and "div curl $=0$".
• Pullback $F^{\ast }:{\Omega }^p(N)\to {\Omega }^p(M)$ along a smooth map $F:M\to N$: substitutes $F$ into a form. It commutes with $d$ ($F^{\ast }d=d{F}^{\ast }$) and with $\wedge$, and — unlike pushforward of vectors — needs no invertibility. Pullback is what makes forms the objects you can integrate.
• Interior product ${\iota }_X:{\Omega }^p\to {\Omega }^{p-1}$: contraction with a vector field $X$. With $d$ it builds the Lie derivative via Cartan's formula ${\mathcal{L}}_X=d{\iota }_X+{\iota }_{X}d$ (see vector-fields-flows.md).

Orientation and integration

A form of top degree $n$ can be integrated over an oriented $n$-manifold. An orientation is a consistent choice of sign for top forms (equivalently a nowhere- vanishing $n$-form, a volume form); not every manifold admits one (the Möbius band does not). Given an orientation, ${\int }_M\omega$ is defined by pulling back to coordinate charts and summing with a partition of unity — coordinate-independent precisely because forms transform by the Jacobian determinant.

The capstone is the generalized Stokes theorem (proved in analysis/stokes-general.md): $${\int }_{M}d\omega ={\int }_{\partial M}\omega .$$ Pairing "$d^2=0$" with "${\partial }^2=0$" (a boundary has no boundary) is exactly what makes de Rham cohomology work.

Forms valued in a vector space

For gauge theory one needs forms valued in a Lie algebra (or a vector space): $\mathfrak{g}$-valued $p$-forms $\omega ={\omega }^a{T}^a$ with each ${\omega }^a$ an ordinary $p$-form and $T^a$ a basis of $\mathfrak{g}$. The wedge combines with the Lie bracket, $$[\alpha ,\beta ]={\alpha }^a\wedge {\beta }^b[T^a,T^b],$$ and this graded bracket is the notation in which the connection 1-form and curvature 2-form of connections.md and curvature.md are written. The gauge field $A=A_{\mu }^a{T}^{a}d{x}^{\mu }$ is precisely a $\mathfrak{g}$-valued 1-form.

References

• Lee, Introduction to Smooth Manifolds, Ch. 11–12, 14, 16.
• Bott & Tu, Differential Forms in Algebraic Topology, Ch. 1.
• Nakahara, Geometry, Topology and Physics, Ch. 5.