In the above normal coordinates, we have the following expansions of the metric tensor $g$ of $Q$ along the normal directions,

(3.18)		$\displaystyle g_{\alpha\beta}|_{(x,y)}$	$\displaystyle=\delta_{\alpha\beta}-\frac{1}{3}\Rm_{\alpha\gamma\xi\beta}\Big|_{(x,0)}y_{\gamma}y_{\xi}+\widetilde{O}(r^{3}),$
(3.19)		$\displaystyle g_{ij}|_{(x,y)}$	$\displaystyle=g^{P}_{ij}(x)+2\IIs^{\alpha}_{ij}\Big|_{(x,0)}y_{\alpha}-(\Rm_{i\gamma\xi j}+\langle\nabla_{\partial_{x_{i}}}\partial_{y_{\gamma}},\nabla_{\partial_{x_{j}}}\partial_{y_{\xi}}\rangle)\Big|_{(x,0)}y_{\gamma}y_{\xi}+\widetilde{O}(r^{3}),$
(3.20)		$\displaystyle g_{i\alpha}|_{(x,y)}$	$\displaystyle=\langle\nabla_{\partial_{x_{i}}}\partial_{y_{\gamma}},\partial_{y_{\alpha}}\rangle\Big|_{(x,0)}y_{\gamma}-\frac{2}{3}\Rm_{i\gamma\xi\alpha}\Big|_{(x,0)}y_{\gamma}y_{\xi}+\widetilde{O}(r^{3}),$

where $g^{P}=(g^{P}_{ij})$ denotes the restriction of the metric $g$ to $U\subset P$, $\Rm$ denotes the Riemann curvature tensor of $g$.