By definition

(3.88)

$$\langle\eta_{\alpha},\eta_{\beta}\rangle=\langle dy_{\alpha}+p_{i\alpha}dx_{i},dy_{\beta}+p_{j\beta}dx_{j}\rangle.$$

By (3.77) we get

(3.89)

$$\langle dy_{\alpha},dy_{\beta}\rangle=\delta_{\alpha\beta}+\widetilde{O}(r^{2}).$$

Also we have $p_{i\alpha}=\widetilde{O}(r)$ and $\langle dx_{i},dy_{\beta}\rangle=\widetilde{O}(r)$ for all $i$ and $\alpha$. The conclusion then follows. ∎