In this exercise Bott and Tu asks to use the decomposition $GL(n,\mathbb R)=O(n)\times \{Positive \ definite\ symmetric \ matrices \} $ to show that the structure group of any real vector bundle can be reduced to $O(n)$. The idea is, given a cocylce of transition maps defining our bundle, to construct an equivalent cocylce with values in $O(n)$. However, if the cocyle is defined over an open cover $(U_\alpha )$, then one must produce appropriate maps $U_\alpha \to GL(n,\mathbb R)$. The only data I have over the $U_\alpha $ is the trivialization of the bundle, to which I can't apply this decomposition.
How can I produce these functions?
Thank you in advance!
I seem to figure out a way where the Riemannian metric on the base space has to be used.
$\forall \alpha$, we fix orthonormal basis sections $\{s^\alpha_1(x),\cdots,s^\alpha_n(x)\}$. Then for each $x$ on the overlap $U_\alpha\cap U_\beta$, $\exists!\ G_{\beta\alpha}(x)\in\mathrm{O}_n(\mathbb{R})$, s.t. $$\big{(} s^\beta_1(x),\cdots,s^\beta_n(x)\big{)}=G_{\beta\alpha}(x)\big{(} s^\alpha_1(x),\cdots,s^\alpha_n(x)\big{)}.$$ Let $\phi_\alpha(x)\big{(} s^\beta_1(x),\cdots,s^\beta_n(x)\big{)}=\lambda_\alpha(x)\big{(} e_1,\cdots,e_n\big{)}$, $\forall\alpha$. So now we have $g_{\beta\alpha}(x)=\phi_\beta(x)\circ\phi^{-1}_\alpha(x)=\lambda_\beta(x)\circ G_{\beta\alpha}(x)\circ\lambda_\alpha(x)^{-1}$.