Subjects: Differential Geometry
Links: The Cotangent Bundle, Covector Fields on Smooth Manifolds, Local and Global Sections of Vector Bundles, Exterior Algebra of Vector Spaces, The Dual Functor and the Multicovector Functor, The Tensor Bundles, Derivations

Def: A section of ${\bigwedge }^k(T^{\ast }M)$ is called a differential $k$-form, or just a $k$-form; this is a continuous tensor field whose value at each point is an alternating tensor. The integer $k$ is called the degree of the form. We denote the space of Smooth $k$-forms by $$\Omega^k(M) := \Gamma\left({\textstyle \bigwedge}^{! k}( T^* M)\right). $$
The wedge product of two differential forms is defined pointwise: $(\omega \wedge \eta {)}_p:={\omega }_p\wedge {\eta }_p$. Thus, the wedge product of a $k$-form with an $l$-form is a $(k+l)$-form. If $f$ is a $0$-form and $\eta$ is a $k$-form, we interpret the wedge product $f\wedge \eta$ to mean the ordinary product $f\eta$. If we define $$\Omega^*(M) := \bigoplus_{k = 0}^n\Omega^k(M),$$then the wedge product turns ${\mathrm{\Omega }}^{\ast }(M)$ into an associative, anticommutative graded algebra.

In any smooth chart, a $k$-form $\omega$ can be written locally as $$\omega = \omega_ I; dx^{i_1}\wedge\dots \wedge dx^{i_k} = \omega_I ; dx^I,$$where the coefficients ${\omega }_I$ are continuous functions defined on the coordinate domain, and use $d{x}^I$ as the abbreviation for $d{x}^{i_1}\wedge \cdots \wedge d{x}^{i_k}$. We are also extending Einstein's summation convention to differential forms, where it is is understood that we are summing over all increasing multi-indices.

Obs: A differential form $\omega$ is smooth on $U$ iff if the component functions ${\omega }_I$ are smooth.

Obs: We can see that $d{x}^I$ are the elementary differential forms, and satisfy the following identity $$dx^{i_1}\wedge\dots \wedge dx^{i_k}\left(\frac{\partial}{\partial x^{j_1}}, \dots,\frac{\partial}{\partial x^{j_k}} \right) = \delta^I_J. $$Thus the component functions ${\omega }_I$ of $\omega$ are determined by $$\omega_I = \omega \left(\frac{\partial}{\partial x^{i_1}}, \dots,\frac{\partial}{\partial x^{i_k}} \right)$$

Def: If $F:M\to N$ is a smooth map and $\omega$ is a differential form on $N$, the pullback $F^{\ast }\omega$ is a differential form on $M$, defined as for any covariant tensor field: $$(F^*\omega)p(v_1,\dots, v_k) = \omega(dF_p (v_1),\dots, dF_p(v_k)). $$

Lemma: Suppose $F:M\to N$ is smooth.

• $F^{\ast }:{\mathrm{\Omega }}^k(N)\to {\mathrm{\Omega }}^k(M)$ is linear over $\mathbb{R}$.
• $F^{\ast }(\omega \wedge \eta )=(F^{\ast }\omega )\wedge (F^{\ast }\eta )$.
• In any smooth chart, $$F^*(\omega_I ; dy^{i_1}\wedge\dots\wedge dy^{i_k}) = (\omega_I \circ F); d(y^{i_1}\circ F)\wedge \dots \wedge d(y^{i_1}\circ F).$$

Pullback Formula for Top-Degree Forms: Let $F:M\to N$ be a smooth map between $n$-manifolds with or without boundary. If $(x^i)$ and $(y^i)$ are smooth coordinates on open subsets $U\subseteq M$ and $V\subseteq N$, respectively, and $u$ is a continuous real-valued function on $V,$then the following golds on $U\cap F^{-1}[V]$: $$F^*(u; dy^1\wedge\dots \wedge dy^n) = (u \circ F)\det\left(\frac{\partial F_i}{\partial x^j}\right) ; dx^1\wedge \dots \wedge dx^n.$$
Cor: If $(U,(x^i))$ and $(V,(y^i))$ are overlapping smooth coordinate charts on $M$. then the following identity holds on $U\cap V$: $$dy^1\wedge \dots \wedge dy^n = \det\left(\frac{\partial y^i}{\partial x^j}\right) ; dx^1\wedge\dots \wedge dx^n.$$
Def: Interior multiplication also extends naturally to vector fields and differential forms, simply by letting it act pointwise: if $X\in \mathfrak{X}(M)$ and $\omega \in {\mathrm{\Omega }}^k(M)$, define a $(k-1)$-form $X⌟\omega =i_X\omega$ by $$(X; \lrcorner; \omega)_p := X_p;\lrcorner; \omega_p. $$
Properties of the Interior Product: Let $X$ be a smooth vector field on $M$.

• If $\omega$ is a smooth differential form, the $i_X\omega$ is smooth.
• $i_X:{\mathrm{\Omega }}^k(M)\to {\mathrm{\Omega }}^{k-1}(M)$ is linear over ${\mathcal{C}}^{\mathrm{\infty }}(M)$ and therefore corresponds to a smooth bundle homomorphism $i_X:{\bigwedge }^k(T^{\ast }M)\to {\bigwedge }^{k-1}(T^{\ast }M)$.

An important operation to differential forms is the exterior derivative

Prop: Suppose $M$ is a smooth manifold and $X\in \mathfrak{X}(M)$. Then the interior multiplication $i_X:{\mathrm{\Omega }}^{\ast }(M)\to {\mathrm{\Omega }}^{\ast }(M)$ is an antiderivation of degree $-1$ whose square is $0$.

Cartan's Lemma: Let $M$ be a smooth $n$-manifold with or without boundary, and let $({\omega }^1,\dots ,{\omega }^k)$ be an ordered $k$-tuple of smooth $1$-forms on an open subset $U\subseteq M$ such that $({\omega }^1{|}_p,\dots ,{\omega }^k{|}_p)$ is linearly independent for each $p\in U$. Given smooth $1$-forms ${\alpha }^1,\dots ,{\alpha }^k$ on $U$ such that $$\sum_{i = 1}^j \alpha^i \wedge \omega^i = 0,$$then each ${\alpha }^i$ can be written as a linear combination of ${\omega }^1,\dots ,{\omega }^k$ with smooth coefficients.

Prop: For each nonnegative integer $k$, there is a contravariant functor ${\mathrm{\Omega }}^k:\mathsf{D}\mathsf{i}\mathsf{f}\mathsf{f}\to \mathsf{V}\mathsf{e}{\mathsf{c}}_{\mathbb{R}}$, which to each smooth manifold $M$ assigns the vector space ${\mathrm{\Omega }}^k(M)$ and to each smooth $F$ the pullback $F^{\ast }$. We see that the exterior derivative is a natural transformation from ${\mathrm{\Omega }}^k$ to ${\mathrm{\Omega }}^{k+1}$.

On Riemannian Manifolds

Let $(M,g)$ be an oriented Riemannian $n$-manifold.

For each $k=1,\cdots n$, $g$ determines a unique inner product on ${\bigwedge }^k(T_p^{\ast }M)$, denoted by $⟨\cdot ,\cdot {⟩}_g$ just like the inner product $T_{p}M$ satisfying $$\langle\omega^1\wedge \dots \wedge \omega^k,\eta\wedge\dots \wedge\eta^k \rangle_k := \det(\langle\omega^i, \eta^j\rangle_g) $$whenever ${\omega }^1,\dots ,{\omega }^k,{\eta }^1,\dots ,{\eta }^k$ are covectors at $k$. We are using The Tangent-Cotangent Bundle Isomorphism for Riemannian Manifolds.

We see that the Riemannian volume form $d{V}_g$ is the unique positively oriented $n$-form that has unit norm with respect this inner product.

Prop: For each $k=0,\dots ,n$ there is a unique smooth bundle homomorphism $\star :{\bigwedge }^k(T^{\ast }M)\to {\bigwedge }^{n-k}(T^{\ast }M)$ satisfying $$\omega\wedge (\star\eta) =\langle \omega, \eta\rangle_g ; dV_g $$for all smooth $k$-forms $\omega$, $\eta$.

Def: The map from the above proposition is called the Hodge star operator.

In any smooth local coordinates $(x^i)$, we can calculate the Hodge dual of a basic $k$-form $$\star(dx^{i_1}\wedge\dots \wedge dx^{i_k}) = \sqrt{\det g} ;g^{i_1 j_1}\cdots g^{i_k j_k} \varepsilon_{j_1,\dots, j_k} ; dx^{j_{k+1}}\wedge \dots\wedge dx^{j_n},$$where ${\epsilon }_{j_1,\dots ,j_n}$ is the Levi-Civita Symbol with ${\epsilon }_{1,\dots ,n}=1$. This we can extend this to general differential forms. Let $\alpha ={\alpha }_{i_1,\dots ,i_k}d{x}^{i_1}\wedge \cdots \wedge d{x}^{i_k}$, $$\star\alpha = \sqrt{\det g} ;g^{i_1 j_1}\cdots g^{i_k j_k} \varepsilon_{j_1,\dots, j_k}; \alpha_{i_1,\dots, i_k} ; dx^{j_{k+1}}\wedge \dots\wedge dx^{j_n}. $$

Prop:

• We see that $\star :{\bigwedge }^0(T^{\ast }M)\to {\bigwedge }^n(T^{\ast }M)$ is given by $\star f=fd{V}_g$.
• We see that $\star (\star \omega )=(-1{)}^{k(n-k)}\omega$ if $\omega$ is a $k$-form.