Due June 7.

Let \(\eta _1, \ldots , \eta _k \in \Omega ^1 (M)\), and \(X_1, \ldots , X_k \in \mathfrak {X}(M)\). Show that the value of
\[ \eta _1 \wedge \cdots \wedge \eta _k ( X_1, \ldots , X_k) \in C^{\infty }(M) \]
at a point \(p \in M\) is the determinant of the \(k \times k\) matrix whose \((i,j)\)-th entry is the number \(\eta _i (X_j) (p)\).
Let \(\omega \in \Omega ^k (M) \), \(p \in M\), and
\[ \{ v_1, \ldots , v_k \} , \{ w_1, \ldots , w_k \} \subset T_pM \]
two sets of vectors that span the same \(k\)-dimensional subspace of \(T_pM\). Write
\[ v_j = \sum _{j = 1}^k a_{ij} w_j . \]
Show that
\[ \omega (p) (v_1, \ldots , v_k ) = \det ( (a_{ij})_{ij}) \omega (p) (w_1, \ldots , w_k). \]
Let \(\eta \in \Omega ^1 (M) \) and \(\omega \in \Omega ^k (M)\).

  • Show that \(d\eta \in \Omega ^2 (M)\). That is, show that

    \begin{gather*} d\eta (fX , Y ) = d\eta (X, fY) = f d\eta (X, Y ) \\ d \eta (X,Y) = - d \eta (Y,X) . \end{gather*}
    for all \(X,Y \in \mathfrak {X}(M)\) and \(f \in C^{\infty }(M)\).

  • Show that

    \[ d \omega : \mathfrak {X}(M) \times \cdots \mathfrak {X}(M) \to C^{\infty } (M) \]
    is \(C^{\infty }(M)\)-multilinear. That is,
    \begin{align*} d\omega (fX_0, \ldots , X_k) & = d\omega (X_0, fX_1, \ldots , X_k) \\ & = \ldots \\ & = d\omega (X_0, \ldots , fX_k) \\ & = f d\omega (X_0 ,\ldots , X_k) \end{align*}

    for all \(X_0, \ldots , X_k \in \mathfrak {X}(M) \) and \(f \in C^{\infty } (M)\). The computation is very short, so explain explicitly each step in words too.

  • Convince yourself (not me!) that \(d\omega \in \Omega ^{k+1}(M)\). That is, if you swap the order of two entries, you get a minus sign:

    \[ d \omega (X_0, \ldots , X_k) = - d\omega (X_0 , \ldots , X_{i+1}, X_i , \ldots , X_k). \]
    This is a mattter of counting signs in the definition of \(d \omega \).

Let \(M\) be an \(n\)-dimensional manifold with smooth structure \(\mathcal {A}\). Show that:

  • For any orientation, there is \(\omega \in \Omega ^n (M)\) that determines the orientation.

  • If \(\omega \in \Omega ^n (M)\) satisfies \(\omega (p) \neq 0\) for all \(p \in M\), then the set

    \[ \{ (U , \varphi ) \in \mathcal {A} \, \vert (\varphi ^{-1} )^{\ast } \omega (\partial _1 , \ldots , \partial _n ) > 0 \, \text { everywhere in } \varphi (U) \} \]
    is an orientation on \(M\).

In particular, \(M\) is orientable if and only if there is a nowhere-vanishing form \(\omega \in \Omega ^n (M)\).

Let \(M\) be a manifold of dimension \(2n +1\). We say that \(\eta \in \Omega ^1 (M)\) is a contact structure if
\[ \eta \wedge \underbrace { d \eta \wedge \cdots \wedge d \eta }_{n \text { times}} \in \Omega ^{2n + 1} (M) \]
is not zero anywhere (in particular, \(M\) is orientable). Let
\[ \eta : = d z + x dy \in \Omega ^1 (\mathbb {R}^3) . \]
Compute \(d\eta \) , \(\eta \wedge d \eta \), and show that \(\eta \) is a contact structure on \(\mathbb {R}^3\).
Let \(M\) be a smooth manifold. The tautological \(1\)-form in the cotangent bundle is the form \(\eta \in \Omega ^1 (T^{\ast }M) \) given by
\[ \eta (\alpha ' (0) ) : = \alpha (0) ( (\pi \circ \alpha )' (0) ) \]
for a curve \(\alpha : (- \varepsilon , \varepsilon ) \to T^{\ast }M \), where \(\pi : T^{\ast } M \to M\) is the natural projection. Show that for any section \(s \in \Gamma (T^{\ast }M)\), one has
\[ s ^{\ast } \eta = s . \]