The above expansions can be proved using the Jacobi fields. Fix a point $q=(0^{k_{0}},x_{1},\ldots,x_{m-k_{0}})\in U\subset P$, we we choose a unit vector $v=\sum\limits_{\alpha=1}^{m-k_{0}}v_{\alpha}\partial_{y_{\alpha}}\in N(q)\cong\mathbb{R}^{k_{0}}$ with $|v|=1$. Let $\vartheta$ be the following radial geodesic in $\mathcal{U}$,

(3.21)

$$\vartheta(t)=\Exp_{P}(q,tv)\equiv\Exp_{q}(tv)$$

such that $\vartheta^{\prime}(0)=v$. In the normal coordinates, the geodesic $\vartheta$ can represented as $\vartheta(t)=(tv_{1},\ldots,tv_{k_{0}},x_{1},\ldots,x_{m-k_{0}})$.

For each $1\leq\alpha\leq k_{0}$ and $1\leq i\leq m-k_{0}$, we define the geodesic variations

(3.22)		$\displaystyle\sigma_{\alpha}(t,s)$	$\displaystyle\equiv((tv_{1},\ldots,t(v_{\alpha}+s),\ldots,tv_{k_{0}},x_{1},\ldots,x_{m-k_{0}}),$
(3.23)		$\displaystyle\sigma_{i}(t,s)$	$\displaystyle\equiv(tv_{1},\ldots,tv_{k_{0}},x_{1},\ldots,x_{i}+s,\ldots x_{m-k_{0}}).$

Then variation fields of $\sigma_{\alpha}(t,s)$ and $\sigma_{i}(t,s)$ give the following Jacobi fields along the radial geodesic $\vartheta(t)$ respectively:

(3.24)

$\displaystyle\begin{cases}J_{\alpha}(t)=t\cdot\partial_{y_{\alpha}},&1\leq\alpha\leq k_{0}\\ J_{i}(t)=\partial_{x_{i}},&1\leq i\leq m-k_{0}.\end{cases}$

By definition,

(3.25)

$$J_{\alpha}(0)=0,\ J_{i}(0)=\partial_{x_{i}}.$$

Taking first derivatives at $t=0$,

(3.26)

$$J_{\alpha}^{\prime}(0)=\partial_{y_{\alpha}},\ J_{i}^{\prime}(0)=v_{\alpha}\nabla_{\partial_{x_{i}}}\partial_{y_{\alpha}}.$$

Then applying the Jacobi equation along the geodesic $\vartheta$,

(3.27)

$\displaystyle\begin{cases}J_{\alpha}^{\prime\prime}+\Rm(J_{\alpha},\vartheta^{\prime})\vartheta^{\prime}=0,\\ J_{i}^{\prime\prime}+\Rm(J_{i},\vartheta^{\prime})\vartheta^{\prime}=0,\\ \end{cases}$

where $\Rm(X,Y)Z\equiv\nabla_{X}\nabla_{Y}Z-\nabla_{Y}\nabla_{X}Z-\nabla_{[X,Y]}Z$ denotes the Riemann curvature tensor of $g$, so it follows that

(3.28)		$\displaystyle J^{\prime\prime}_{\alpha}(0)=0,\ J^{\prime\prime\prime}_{\alpha}(0)=-v_{\gamma}v_{\xi}\Rm(\partial_{y_{\alpha}},\partial_{y_{\gamma}})\partial_{y_{\xi}},$
(3.29)		$\displaystyle J^{\prime\prime}_{i}(0)=-v_{\gamma}v_{\xi}\Rm(\partial_{x_{i}},\partial_{y_{\gamma}})\partial_{y_{\xi}}.$

Therefore,

(3.30)		$\displaystyle g_{\alpha\beta}$	$\displaystyle=t^{-2}g(J_{\alpha},J_{\beta})=\delta_{\alpha\beta}-\frac{1}{3}\Rm_{\alpha\gamma\xi\beta}v_{\gamma}v_{\xi}t^{2}+\widetilde{O}(t^{3}),$
(3.31)		$\displaystyle g_{ij}$	$\displaystyle=g(J_{i},J_{j})=g^{P}_{ij}+2v_{\alpha}\IIs_{ij}^{\alpha}t-\Big(\Rm_{i\gamma\xi j}+\langle\nabla_{\partial_{x_{i}}}\partial_{y_{\gamma}},\nabla_{\partial_{x_{j}}}\partial_{y_{\xi}}\rangle\Big)v_{\gamma}v_{\xi}t^{2}+\widetilde{O}(t^{3}),$
(3.32)		$\displaystyle g_{i\alpha}$	$\displaystyle=t^{-1}g(J_{i},J_{\alpha})=\langle\nabla_{\partial_{x_{i}}}\partial_{y_{\gamma}},\partial_{y_{\alpha}}\rangle v_{\gamma}t-\frac{2}{3}\Rm_{i\gamma\xi\alpha}v_{\gamma}v_{\xi}t^{2}+\widetilde{O}(t^{3}).$

Let $y_{\alpha}=tv_{\alpha}$, then we obtain the desired expansions. ∎