Differential Forms

Vector Bundles

Using §Tangent and Cotangent Bundles, ⁋Example 5 and §Tangent and Cotangent Bundles, ⁋Theorem 6, we can define the following.

Definition 1 For a manifold \(M\),

\[\mathcal{T}^{r,s}(M)=\mathcal{T}^{r,s}(TM),\quad \bigwedge\nolimits^\ast(M)=\bigwedge(T^\ast M),\quad \bigwedge\nolimits^k(M)=\bigwedge\nolimits^k(T^\ast M)\]

are called the \((r,s)\)-tensor bundle, exterior algebra bundle, and exterior \(k\)-bundle over \(M\), respectively. The elements of their smooth sections

\[\Gamma\left(\mathcal{T}^{r,s}(M)\right),\quad\Omega^\ast(M):=\Gamma\left(\bigwedge\nolimits^\ast(M)\right),\quad\Omega^k(M):=\Gamma\left(\bigwedge\nolimits^k(M)\right)\]

are called a tensor field, a differential form, and a differential \(k\)-form, respectively.

For two simple tensors

\[\omega=\alpha^1\otimes\cdots\otimes \alpha^r\otimes u_{r+1}\otimes\cdots\otimes u_{r+s}\in\mathcal{T}^{r,s}(T_p^\ast M),\quad u=u_1\otimes\cdots\otimes u_r\otimes \alpha^{r+1}\otimes\cdots\otimes \alpha^{r+s}\in\mathcal{T}^{r,s}(T_pM)\]

define

\[(\omega,u)=\alpha^1(u_1)\alpha^2(u_2)\cdots \alpha^{r+s}(u_{r+s})\]

Then, since \((-,-)\) is a non-degenerate pairing, \(\mathcal{T}^{r,s}(T_p^\ast M)\cong\mathcal{T}^{r,s}(T_pM)^\ast\) holds. ([Linear Algebra] §Dual Space, ⁋Corollary 5)

Similarly, for two elements

\[\omega=\alpha^1\wedge\cdots\wedge \alpha^k\in \bigwedge\nolimits^k(T_p^\ast M),\quad u=u_1\wedge\cdots\wedge u_k\in\bigwedge\nolimits^k(T_pM)\]

if we define the pairing \((-,-)\) by

\[(\omega, u)=\det\bigl(\alpha^i(u_j)\bigr)\]

we can verify that \(\bigwedge\nolimits^k(T_pM)^\ast\cong\bigwedge\nolimits^k(T_p^\ast M)\). On the other hand, for a finite family of vector spaces \((V_i)_{1\leq i\leq n}\),

\[\bigoplus_{i=1}^n V_i^\ast\cong \left(\bigoplus_{i=1}^n V_i\right)^\ast\]

holds, and since \(\bigwedge(V)\) is a direct sum of only finitely many \(\bigwedge\nolimits^k(V)\),

\[\bigwedge(T_p^\ast M)=\bigoplus_{k\geq 0}\bigwedge\nolimits^k(T_p^\ast M)\cong\bigoplus_{k\geq 0}\bigwedge\nolimits^k(T_pM)^\ast\cong\left(\bigwedge(T_pM)\right)^\ast\]

holds.

Differential Forms and Pullbacks

Among the objects in Definition 1 above, the elements of \(\Omega^\ast(M)\) are of particular interest. By definition, any differential form \(\omega\in\Omega^\ast(M)\) is a map \(M\rightarrow\bigwedge\nolimits^\ast(M)\), and this value is written as

\[p\mapsto \omega_p\in\bigwedge(T_p^\ast M)\]

If we define the wedge product \(\omega\wedge\eta\) of two differential forms by the formula

\[(\omega\wedge\eta)_p=\omega_p\wedge\eta_p\qquad\text{for all $p\in M$}\]

then, letting \(n=\dim M\), we can regard \(\Omega^\ast(M)\) as the \(\mathbb{N}\)-graded \(\mathbb{R}\)-algebra

\[\Omega^\ast(M)=\bigoplus_{k=0}^n\Omega^k(M)\]

Furthermore, because the scalar multiplication on \(\Omega^\ast(M)\) by \(\mathbb{R}\) can actually be performed at each point \(p\), we may also view the coefficients of \(\Omega^\ast(M)\) as \(C^\infty(M)\). Algebraically, this can be viewed as changing the coefficient ring via the ring homomorphism \(\mathbb{R}\rightarrow C^\infty(M)\), and henceforth we always regard \(\Omega^\ast(M)\) as being endowed with this \(\mathbb{N}\)-graded \(C^\infty(M)\)-algebra structure.

Now suppose a \(C^\infty\) map \(F:M\rightarrow N\) is given. Then the linear map \(\dd{F_p}:T_pM\rightarrow T_{F(p)}N\) is well-defined. Therefore, applying the functoriality of the exterior algebra to the dual map of \(\dd{F_p}\), we obtain

\[\bigwedge({\dd{F}}_p^\ast):\bigwedge(T_{F(p)}^\ast N)\rightarrow\bigwedge(T_p^\ast M)\]

([Multilinear Algebra] §Tensor Algebra, ⁋Proposition 11) At each point \(p\), assigning \(\bigwedge({\dd{F}}_p^\ast)\) gives a linear map \(\Omega^\ast(N)\rightarrow\Omega^\ast(M)\), which we denote by \(F^\ast\). That is, for any \(\omega\in\Omega^\ast(N)\),

\[(F^\ast\omega)_p=\bigwedge({\dd{F}}_p^\ast)(\omega_{F(p)})\]

The differential form \(F^\ast\omega\) obtained in this way is called the pullback of \(\omega\) by \(F\). Furthermore, by definition \(F^\ast\) is a graded algebra homomorphism, so it also preserves \(\wedge\).

In particular, suppose \(\omega\) is a \(k\)-form. At a point \(p\in M\), to compute \((F^\ast\omega)_p\), substituting \(k\) vectors \(X_1(p),\ldots, X_k(p)\), we obtain

\[(F^\ast\omega)_p(X_1(p),\ldots, X_k(p))=(F^\ast_p\omega_{F(p)})\bigl(X_1(p),\ldots, X_k(p)\bigr)=\omega_{F(p)}\bigl(\dd{F_p}(X_1(p)), \ldots, \dd{F_p}(X_k(p))\bigr)\]

Exterior Derivative and de Rham Cohomology

At each point \(p\), since \(\bigwedge\nolimits^0(T_p^\ast M)=\mathbb{R}\), \(\Omega^0(M)\) in Definition 1 is equal to \(C^\infty(M)\). For any \(f\in C^\infty(M)\), its differential \(\dd{f}\) is the function that takes each point \(p\in M\) and outputs \(\dd{f_p}:T_pM\rightarrow\mathbb{R}\). (§Examples of Differentials, ⁋Definition 6) That is, \(\dd{f}\in\Gamma(T^\ast M)=\Omega^1(M)\). This operator \(d\) is also defined for general differential forms as follows.

Theorem 2 For a manifold \(M\), there uniquely exists a degree \(1\) anti-derivation \(d:\Omega^\ast(M)\rightarrow\Omega^\ast(M)\) satisfying the following two conditions. (For the proof, see [War].)

1. \(d^2=0\),
2. For any \(f\in\Omega^0(M)\), \(\dd{f}\) is equal to the differential of \(f\) as above.

Furthermore, \(d\) defined in this way commutes with the pullback \(F^\ast\).

A graded algebra equipped with such a differential \(d\) is called a differential graded algebra, or simply a DG-algebra. Meanwhile, by condition 1 above, the following sequence

\[0\longrightarrow\Omega^0(M)\overset{d}{\longrightarrow}\Omega^1(M)\overset{d}{\longrightarrow}\Omega^2(M)\overset{d}{\longrightarrow}\cdots\overset{d}{\longrightarrow}\Omega^n(M)\longrightarrow 0\tag{1}\]

becomes a cochain complex. In addition, since \(d\) commutes with \(F^\ast\) and \(F^\ast\) is a graded algebra homomorphism, in the above language we can say that \(F^\ast\) induces a chain map between the de Rham complexes.

We call the cohomology group corresponding to the cochain complex of (1) the de Rham cohomology group and denote it by \(H^\ast_\text{dR}(M)\). The de Rham theorem shows that \(H_\text{dR}^\ast(M)\) obtained in this way contains the same information as other topologically defined cohomology groups.

Interior multiplication

Definition 3 Consider a manifold \(M\) and a vector field \(X\) given on it. Then \(\iota_X:\Omega^\ast(M) \rightarrow\Omega^\ast(M)\) is the map that assigns to each \(k\)-form \(\omega\) the \((k-1)\)-form \(\iota_X\omega\) defined by the formula

\[(\iota_X\omega)(X_1,\ldots, X_{k-1})=\omega(X,X_1,\ldots, X_{k-1})\]

where in the case \(k=0\), we set \(\iota_X\omega=0\). This is called the interior multiplication by \(X\).

Proposition 4 For a manifold \(M\) and any vector field \(X\) given on it, the interior multiplication \(\iota_X\) is an antiderivation of degree \(-1\).

---

References

[War] Frank W. Warner. Foundations of Differentiable Manifolds and Lie Groups, Graduate texts in mathematics, Springer, 2013
[Lee] John M. Lee. Introduction to Smooth Manifolds, Graduate texts in mathematics, Springer, 2012

---