Differential Forms

This post was machine-translated from the Korean original by Marvin (via Kimi). It may contain errors or awkward phrasing — the Korean original is the source of truth.

Vector Bundles

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

Definition 1 For a manifold \(M\), we define

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

and call them the \((r,s)\)-tensor bundle, exterior algebra bundle, and exterior \(k\)-bundle on \(M\), respectively. The smooth sections of these bundles,

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

are called tensor fields, differential forms, and differential \(k\)-forms, respectively.

For two simple tensors

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

we define

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

Then \((-,-)\) is a non-degenerate pairing, so \(\mathcal{T}^{r,s}(T_p^\ast M)\cong\mathcal{T}^{r,s}(T_pM)^\ast\). (§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

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

then 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 the direct sum of only finitely many \(\bigwedge\nolimits^k(V)\),

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

holds.

Differential Forms and Pullback

Among the objects introduced in Definition 1, the elements of \(\Omega^\ast(M)\) are of particular interest. By definition, any differential form \(\omega\in\Omega^\ast(M)\) is a function \(M\rightarrow\bigwedge\nolimits^\ast(M)\), and we write its values as

\[p\mapsto \omega_p\in\bigwedge\nolimits^\ast(T_pM).\]

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

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

then \(\Omega^\ast(M)\) can be regarded as an \(\mathbb{N}\)-graded \(\mathbb{R}\)-algebra

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

Moreover, since scalar multiplication in \(\Omega^\ast(M)\) by \(\mathbb{R}\) can in fact be performed at each point \(p\), we may also regard the coefficients of \(\Omega^\ast(M)\) as \(C^\infty(M)\). Algebraically, this can be thought of as changing the coefficient ring via the ring homomorphism \(\mathbb{R}\rightarrow C^\infty(M)\), and henceforth we always consider \(\Omega^\ast(M)\) as an \(\mathbb{N}\)-graded \(C^\infty(M)\)-algebra in this manner.

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

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

(§Tensor Algebras, ⁋Definition 10) Let \(F^\ast\) denote the linear map \(\Omega^\ast(N)\rightarrow\Omega^\ast(M)\) obtained by assigning \(\bigwedge({dF}_p^\ast)\) to each point \(p\). That is, for any \(\omega\in\Omega^\ast(N)\),

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

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

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

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

Exterior Derivative and de Rham Cohomology

Earlier we verified that \(\Omega^0(M)=C^\infty(M)\). For any \(f\in C^\infty(M)\), its differential \(df\) is the function that takes each point \(p\in M\) and outputs \(df_p:T_pM\rightarrow\mathbb{R}\). (§Examples of Differentials, ⁋Definition 6) That is, \(df\in T^\ast M=\Omega^1(M)\). This operator \(d\) is defined for general differential forms as follows.

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

1. \(d^2=0\),
2. For any \(f\in\Omega^0(M)\), \(df\) coincides with the differential of \(f\) as above.

Moreover, this \(d\) commutes with 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 sequence

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

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

We call the cohomology group of the cochain complex (2) the de Rham cohomology group and denote it by \(H^\ast_\text{dR}(M)\). de Rham’s theorem shows that \(H_\text{dR}^\ast(M)\) obtained in this way carries the same information as other cohomology groups defined topologically.

Interior Multiplication

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

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

This is called interior multiplication by \(X\).

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

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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

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