Differential Forms

This post was translated from Korean by LLM (Kimi). The translation may contain errors or awkward sentences. 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 \(\mathcal{T}^{r,s}(T_p^\ast M)\cong\mathcal{T}^{r,s}(T_pM)^\ast\). ([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

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

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

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

([Multilinear Algebra] §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. (##ref##)

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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