# Vector subbundle and frame field relation

Question:

Let $$E \to M$$ be a vector bundle of rank $$k$$. Suppose that for each $$p \in M$$ we are given a subspace $$E'_p$$ of $$E_p$$ and consider the set $$\displaystyle E' = \bigcup_{p \in M} E'_p$$.

Show that $$E'$$ is the total space of a rank $$l$$ subbundle if and only if for each $$p\in M$$ there is an open set $$U$$ of $$p$$ on which smooth sections $$\sigma_1, \ldots, \sigma_l$$ are defined such that for each $$q \in M$$ the set $$\{\sigma_1(q), \ldots, \sigma_l(q)\}$$ is a basis of the subspace $$E'_q$$.

Attempt of proof:

($$\implies$$) Suppose $$E'$$ is the total space of a rank $$l$$ subbundle then for every $$p \in M$$ there exists a VB-chart such that

$$\phi (\pi^{-1}(U) \,\,\cap\,\,E' ) = U \times V' \subseteq U \times V$$

This rises a chart of the triple $$(E', \pi|_{E'}, M, V')$$ if we define the $$\phi'$$ as $$\phi|_{\pi^{-1}_{E'}(U)}$$ (the restriction of $$\phi$$ to $$\pi^{-1}_{E'}(U)$$), where $$\pi^{-1}(U) \cap E' = \pi^{-1}|_{E'}(U)$$ and such that the following diagram commutes

$$\require {AMScd} \begin{CD}\pi^{-1}(U) \cap E' = \pi^{-1}|_{E'}(U) @>\phi'>> U \times V' \subseteq U \times V\\@V \pi|_{E'} VV @VVpr_1V\\U @>>id >U\end{CD}$$

Let $$\{e_1, \ldots, e_l\}$$ be a basis of $$V'$$. Since $$\phi$$ is a diffeomorphism we maydefine the sections of $$E'$$ as

$$\sigma_j (q) = \phi^{-1} (q, e_j) \, \, , j = 1, \ldots, l , \forall q \in U$$

Now the inclusion $$\iota_j : U \to U \times V'$$, $$\iota_j (x) = (x, e_j)$$ is a section of $$U \times V'$$ since $$pr_1 \circ \iota_j = id_U$$ and as $$\phi'$$ is a bundle isomorphism then $$\sigma_j$$ are sections of $$E'$$. Since the vectors $$(q, e_j)$$ form a basis of $$\{q\}\times V'$$ and $$\phi'$$ is an isomorphism then $$\{\sigma_1(q), \ldots, \sigma_l(q)\}$$ is a basis of $$E'_q$$, ( notice that $$\sigma_j(q) \in E'_q, \forall j$$) for each $$q$$ as wanted.

($$\Longleftarrow$$) Suppose there exists an open neighborhood $$U$$ of $$p$$ (taken arbitrarily) such that for each $$q \in U$$ the set $$\{\sigma_1(q), \ldots, \sigma_l(q)\}$$ is a basis of the subspace $$E'_q$$.

Define $$\phi' : U \times V' \to E' \cap \pi^{-1}(U)$$ by $$\phi' (q, c_1, \ldots, c_l) = \displaystyle \sum_{j=1}^l c^j\sigma_j(q)$$. As $$\sigma_j$$ are smooth then $$\phi$$ is smooth.

We have that $$\pi(\phi'(q,c)) = q = pr_1 (q,c)$$ and the restriction of $$\phi'$$ to $$\{q\} \times V'$$ is linear. Since $$\sigma_1(q), \ldots, \sigma_l(q)$$ are linearly independent the homomorphism $$\phi'$$ is injective and thus an isomorphism on each fiber.

We conclude that $$\phi$$ is a bundle isomorphism and $$\phi'^{-1}(E' \cap \pi^{-1}(U)) = U \times V'$$

and therefore $$E'$$ is a total space of a rank $$l$$ subbundle.

• I would like to check if this is correct.

• Do I have to show that $$E'$$ is a submanifold of $$E$$?

The proof basically looks fine to me. However, in the second part, you have not really shown that $E'$ is a smooth subbundle of $E$ but only that $E'$ is a vector bundle of rank $\ell$.
To prove that $E'$ is a subbundle, you should complete the sections $\sigma_1,\dots,\sigma_\ell$ to a local frame for $E$. To do this, just use a vector bundle chart $\psi$ for $E$ and for some point $x$, complete $\psi(\sigma_1(x)),\dots,\psi(\sigma_\ell(x))$ to a basis of $V$, say using vectors $v_{\ell+1},\dots, v_n$. Then for $i=\ell+1,\dots,n$ define $\sigma_i(y)$ as $\psi^{-1}(y,v_i)$ on the domain of $\iota$. Then you obtain local smooth sections $\sigma_1,\dots,\sigma_n$ whose values at $x$ form a basis for the fiber $E_x$. Hence there is an open neighbourhood $U$ of $x$ such that the values in each $y\in U$ form a basis for $E_y$. Then as in your proof you can construct a vector bundle chart for $E$ on $U$ in which $E'$ corresponds to a linear subspace.
• This $\psi$ composition seems to be incorrect. I guess it should be $(pr_2 \circ \psi) (\sigma_i (x))$. Commented Jun 17, 2016 at 18:26
• Yes you are right, it should be the second component of $\psi$. Commented Jun 18, 2016 at 10:43
• I've completed the basis of $E'_q$ to $E_q$ (dimension $k$), already using smooth section (this can be done locally) then defined $\psi$ as $\phi^{-1}$ where $$\phi (q,c) = \sum_{j=1}^k c^j \sigma_j(q)$$ this seems to solve the question. Commented Jun 18, 2016 at 10:46