Vector bundles, trivializations, basic constructions, basic properties.

A reference for this material is Chapter 10 of John M. Lee. Introduction to smooth manifolds. Second edition. Grad. Texts in Math., 218. Springer, New York, 2013. xvi+708pp. ISBN: 978-1-4419-9981-8.

Solution:

We do the one of direct sum. The other two are similar, but the computations are a little more involved. Start by defining the set

\[ E \oplus F : = \bigsqcup _{x \in M} (E _x \oplus F_x) \]
and equip it with the map \(\pi _{\oplus } : E \oplus F \to M\) that sends \((E _x \oplus F_x) \) to \(x\). We now construct trivializations using the ones of \(E\) and \(F\).

Take local trivializations \((U , \psi )\) and \((U , \zeta )\) of \(E \) and \(F\), respectively. That is, \(U \subset M\) is an open set, and

\begin{gather*} \psi : \pi _E^{-1} (U) \to U \times \mathbb {R}^k \\ \zeta : \pi _F^{-1} (U) \to U \times \mathbb {R}^{\ell } \end{gather*}
diffeomorphisms, such that for each \(x \in U\), the restrictions \(\psi : E_x \to \{ x \} \times \mathbb {R}^k\) and \(\zeta : F_ x \to \{ x \} \times \mathbb {R}^{\ell }\) are linear isomorphisms. Then define
\[ \psi \oplus \zeta : \pi _{\oplus }^{-1} (U) \to U \times \mathbb {R} ^k \times \mathbb {R}^{\ell } \]
as
\[ ( \psi \oplus \zeta )( v, w ) : = (x , \psi _2 (v), \zeta _2 (w) ) , \]
where \(v \in E_x\), \(w \in F_x\), and \(\psi _2\), \(\zeta _2\) are the second coordinates of \(\psi \) and \(\zeta \), respectively. For each \(x \in U\), the map \(\psi \oplus \zeta : (E _x \oplus F_x) \to \{ x \} \times \mathbb {R}^{k + \ell }\) is a linear isomorphism. However, we need to check that when we consider other trivialization, the change of coordinates is smooth.

Take \((V \overline {\psi }) \), \((V, \overline {\zeta })\) two local trivializiations of \(E\) and \(F\). We need to check that the change of coordinates

\[ (\psi \oplus \zeta ) \circ (\overline {\psi } \oplus \overline {\zeta })^{-1} : (\overline {\psi } \oplus \overline {\zeta }) (\pi _{\oplus } ^{-1} (U \cap V) ) \to ( \psi \oplus \zeta ) (\pi _{\oplus } ^{-1} (U \cap V) ) \]
is smooth. This happens to be very easy. Note that
\[ (\overline {\psi } \oplus \overline {\zeta }) (\pi _{\oplus } ^{-1} (U \cap V) ) = ( \psi \oplus \zeta ) (\pi _{\oplus } ^{-1} (U \cap V) ) = (U \cap V) \times \mathbb {R}^k \times \mathbb {R}^{\ell } . \]
Then
\begin{align*} (\psi \oplus \zeta ) & \circ (\overline {\psi } \oplus \overline {\zeta })^{-1} ( x, \lambda _1, \ldots , \lambda _k , \mu _1, \ldots , \mu _{\ell } ) \\ & = (\psi \oplus \zeta ) ( \overline {\psi } ^{-1} (x, \lambda _1, \ldots , \lambda _k), \overline {\zeta }^{-1}(x, \mu _1, \ldots , \mu _{\ell })) \\ & = ( x , \psi _2 ( \overline {\psi }^{-1} (x, \lambda _1, \ldots , \lambda _k) ) , \zeta _2 (\overline { \zeta }^{-1} ( x, \mu _1, \ldots , \mu _ {\ell } ) ) ) . \end{align*}

Since \(\psi \) and \(\overline {\psi }\) are diffeomorphisms, the second coordinate depends smoothly on \(x\) and the \(\lambda _i\)’s. Since \(\zeta \) and \(\overline {\zeta }\) are diffeomorphisms, the third coordinate depends smoothly on \(x\) and the \(\mu _i \)’s.

Solution:

Start by defining the set

\[ E^{\ast } : = \bigsqcup _{x \in M} (E_x)^{\ast } \]
and equip it with the map \(\pi _{E^{\ast }}: E ^{\ast } \to M\) that sends \((E_x)^{\ast }\) to \(x\). We now construct trivializations using the ones of \(E\).

Let \((U, \psi ) \) be a local trivialization of \(E\). That is, \(U \subset M\) is an open set and \(\psi : \pi _E ^{-1} (U ) \to U \times \mathbb {R}^k\) a diffeomorphism where \(\psi : E_x \to \{ x \} \times \mathbb {R}^k\) is a linear isomorphism for all \(x \in U\). For each \(x \in U\), define \(v_i (x) = \psi ^{-1} (x,e_i)\), where \(e_i = (0, \ldots , 1, \ldots , 0)\) is the \(i\)-th canonical vector. Let \(\{ \eta ^1 (x), \ldots , \eta ^k(x) \} \subset E_x^{\ast }\) be the dual basis.

We can then define \(\psi ^{\ast } : \pi _{E^{\ast }}^{-1}(U) \to U \times \mathbb {R}^k\) by

\[ \psi ^{\ast } \Big ( \sum a_i \eta ^i (x) \Big ) = ( x , a_1, \ldots , a_k ). \]
For each \(x \in U\), the map \(\psi ^{\ast } : E_x^{\ast } \to \{ x \} \times \mathbb {R}^k\) is a linear isomorphism. However, we need to check that when we consider other trivialization, the change of coordinates is smooth.

Consider other local trivialization \((V, \zeta )\). We need to check that

\[\psi ^{\ast } \circ ( \zeta ^{\ast } ) ^{-1} : \zeta ^{\ast } ( \pi _{E^{\ast }}^{-1} (U \cap V) ) \to \psi ^{\ast } (\pi _{E^{\ast }}^{-1} (U \cap V) ) ) \]
is smooth. Note that
\[ \zeta ^{\ast } ( \pi _{E^{\ast }}^{-1} (U \cap V) ) = \psi ^{\ast } (\pi _{E^{\ast }}^{-1} (U \cap V) ) ) = (U \cap V ) \times \mathbb {R}^k . \]
Let \(w_j (x) : = \zeta ^{-1} (x,e_j)\) and \(\{ \alpha ^1 (x) , \ldots , \alpha ^k (x) \} \subset E_x ^{\ast }\) the dual basis. For each \(x \in U \cap V\), find a matrix \(M (x) = (m_{ij} (x))_{ij}\) such that
\[ w_j (x) = \sum _{i} m_{ij}(x) v_i (x) . \]
Since \(\psi \) and \(\zeta \) are diffeomorphisms, the change of charts is smooth, and each \(m_{ij} : U \to \mathbb {R}\) is smooth. If we denote the inverse matrix by \(M^{-1}(x) = (m^{ij}(x))_{ij}\), we have (ommiting the dependence on \(x\)):
\[ \sum _{i} m^{a i } \eta ^{i} (w_{b}) = \sum _{i , \ell } m^{a i } \eta ^{i} (m_{\ell b} v_{\ell } ) = \sum _i m^{ai} m_{ib} = \delta ^a_b , \]
where \(\delta ^a _b \) equals \(1\) if \(a = b\) and \(0\) if \(a \neq b\). Therefore
\[ \alpha ^a = \sum _i m^{ai}\eta ^i . \]
Consequently,
\begin{align*} \psi ^{\ast } ( (\zeta ^{-\ast }&) ^{-1} (x, \lambda _1 , \ldots , \lambda _k ) ) = \psi ^{\ast } \left ( \sum _i \lambda _i \alpha ^i (x) \right ) \\ & = \psi ^{\ast } \left ( \sum _{i, \ell } \lambda _i m^{i\ell } \eta ^{\ell } (x) \right ) \\ & = \left ( x, \sum _{i } \lambda _i m^{i 1} , \ldots , \sum _i \lambda _i m^{ik} \right ). \end{align*}

Since the matrix \(M^{-1}\) depends smoothly on \(x\), the last expression depends smoothly on \(x\) and the \(\lambda _i\)’s.

Solution:

See for example Conlon Lawrence. Differentiable manifolds, a first course. Birkhäuser, 1993.