5 Differential Forms

Definition 5.1 (Differential) If \(f \in C^\infty_p(M)\), then its differential at \(p\) is defined to be the covector \(df \in T_pM^\vee\) such that \[ (df)_p(X) = X(f) \]

Theorem 5.1 At any point \(p \in M\), if \((U, x^1, \dots, x^n)\) is a chart about \(p\), then the differentials \[ dx^1, \quad \dots, \quad dx^n \] comprise a basis for \(T_pM^\vee\) such that \[ df = \sum \frac{\partial f}{\partial x^i} dx^i \] for any \(f \in C_p^\infty(M)\)

5.1 Exterior Power of Cotangent Bundle

Definition 5.2 (Cotangent Bundle) The cotangent bundle of a manifold \(M\) is denoted by \(T^*M\), defined in similar fashion of tangent bundle.

Definition 5.3 (Alternating tensor) A \(k\)-tensor on a vector space \(V\) is defined as a \(k\)-linear function \[ f: V \times \dots \times V \to \mathbb R \] A \(0\)-tensor is simply defined as a scalar in \(\mathbb R\).

A tensor is alternating if \[ f(v_{\sigma(1)}, \dots, v_{\sigma(n)}) = \mathop{\mathrm{sgn}}(\sigma) f(v_1, \dots, v_n) \] where \(\sigma\) is a permutation.

Definition 5.4 (Exterior power of vector space) Let \(V\) be a vector space, then the set of all alternating \(k\)-tensors of \(V\) is denoted with \(A_k(V)\) \[ \begin{aligned} A_0(V) &= \mathbb R \\ A_1(V) &= V^\vee \\ \dots \end{aligned} \]

Sometimes we also denote \(A_k(V) = \bigwedge^k(V^\vee)\), and refer to the space as the \(k\)-th exterior power of \(V^\vee\).

Definition 5.5 (Exterior power of the cotangent bundle) The disjoint union of all \(k\)-th exterior power of all cotangent spaces of a manifold is called the \(k\)-th exterior power of the manifold’s cotangent bundle. \[ \sideset{}{^k}\bigwedge(T^*M) = \bigsqcup_{p \in M} \sideset{}{^k}\bigwedge(T_p^*M) \\ \]

Theorem 5.2 At any point \(p \in M\), if \((U, x^1, \dots, x^n)\) is a chart about \(p\), the collection is a basis for the vector space \(\bigwedge^k(T_p^*M)\) \[ dx^I = dx^{i_1} \wedge \dots \wedge dx^{i_k} \] where \(I = \{i_1, \dots, i_k\}\) samples from \(1, \dots, n\) without replacement. That is, if \(\omega \in \bigwedge^k(T_p^*M)\) then \[ \alpha = \sum \alpha \left (\frac{\partial}{\partial x^I}\right ) dx^I \] where \[ \frac{\partial}{\partial x^I} = \left (\frac{\partial}{\partial x^{i_1}}, \quad \dots, \quad \frac{\partial}{\partial x^{i_n}} \right ) \]

Proposition 5.1 The \(k\)-th exterior power of the cotangent bundle can be defined as a smooth manifold in the similar fashion as tangent bundle.

5.2 Differential Forms

Definition 5.6 (Differential forms) A differential \(k\)-form \[ \omega: M \to \sideset{}{^k}\bigwedge(T^*M) \] is a section of the vector bundle.

Proposition 5.2 If \(\omega\) is a \(k\)-form defined in a neighborhood of \(p\), and written in local coordinates about a point \(p\), \[ \omega = \sum a_I dx^I \] then \(\omega\) is smooth at \(p\) iff all \(a^I\) is smooth at \(p\).

Definition 5.7 (Pull back) If \(F: U \subset M \to N\) is a function defined about \(p\), and \(\alpha \in \bigwedge^k(T_p^*N)\), its pull back \(F_*\alpha \in \bigwedge^k(T_p^*M)\) is defined as \[ F^*\alpha(X_1, \dots, X_n) = \alpha(F_*X_1, \dots, F_*X_n) \]

If \(F \in C^\infty_p(M, N)\) is a smooth function, \(\omega \in C^\infty(N, \bigwedge^k(T_p^*N))\) is a \(k\)-form, then its pull back is defined pointwise \[ (F^*\omega)_p = F^*(\omega_p) \] then \(F^*\omega_p \in C^\infty(M, \bigwedge^k(T_p^*M))\) (see 5.3).

The pull back of a real function \(h\) (\(0\)-form) is the \(0\)-form \[ F^*h = h \circ F \]

Definition 5.8 (Codifferential) If \(F \in C^\infty_p(M, N)\), the codifferential is its pull back of a \(1\)-form, or equivalently, the dual of its differential, or equivalently, \[ F^*(\omega)(X) = \omega(F_* X) \]

Proposition 5.3

If \(F \in C^\infty_p(M, N)\), then for any \(h \in C^\infty_p(N)\), \[ F^*(dh) = d(F^*h) \]
If \(F \in C^\infty_p(M, N)\), \(\omega \in C^\infty_p(M, \bigwedge^r(T^*_M)), \tau \in C^\infty_p(M, \bigwedge^k(T^*_M))\) then their wedge product \(\omega \wedge \tau \in C^\infty_p(M, \bigwedge^{r + k}(T^*_M))\) and \[ F^*(\omega \wedge \tau) = F^*\omega \wedge F^*\tau \]

This proves the pull back of a smooth form is smooth.