We write the full matrix expression of the metric $g$ as

(3.53)

$$g=\left[{\begin{array}[]{cc}g_{\alpha\beta}&0\\ 0&g_{ij}\\ \end{array}}\right]+\left[{\begin{array}[]{cc}0&S\\ S^{t}&0\\ \end{array}}\right],$$

where $S=(g_{\alpha i})=\widetilde{O}(r)$. We denote by $(h_{\alpha\beta})$ and $(h_{ij})$ the inverse matrix of $(g_{\alpha\beta})$ and $(g_{ij})$ respectively. Then by elementary consideration

(3.74)		$\displaystyle g^{-1}$	$\displaystyle=$	$\displaystyle\left[{\begin{array}[]{cc}h_{\alpha\beta}&0\\ 0&h_{ij}\\ \end{array}}\right]-\left[{\begin{array}[]{cc}h_{\alpha\beta}&0\\ 0&h_{ij}\\ \end{array}}\right]\left[{\begin{array}[]{cc}0&S\\ S^{t}&0\\ \end{array}}\right]\left[{\begin{array}[]{cc}h_{\alpha\beta}&0\\ 0&h_{ij}\\ \end{array}}\right]$
			$\displaystyle+$	$\displaystyle\left[{\begin{array}[]{cc}h_{\alpha\beta}&0\\ 0&h_{ij}\\ \end{array}}\right]\left[{\begin{array}[]{cc}0&S\\ S^{t}&0\\ \end{array}}\right]\left[{\begin{array}[]{cc}h_{\alpha\beta}&0\\ 0&h_{ij}\\ \end{array}}\right]\left[{\begin{array}[]{cc}0&S\\ S^{t}&0\\ \end{array}}\right]\left[{\begin{array}[]{cc}h_{\alpha\beta}&0\\ 0&h_{ij}\\ \end{array}}\right]$
			$\displaystyle+$	$\displaystyle\widetilde{O}(r^{3}).$

Notice the third term does not have off-diagonal contributions, so the inverse matrix $g^{-1}=(g^{IJ})$ satisfies

(3.75)		$\displaystyle g^{i\alpha}$	$\displaystyle=-h_{ij}g_{j\beta}h_{\beta\alpha}+\widetilde{O}(r^{3})$
(3.76)		$\displaystyle g^{ij}$	$\displaystyle=h_{ij}+\widetilde{O}(r^{2}),$
(3.77)		$\displaystyle g^{\alpha\beta}$	$\displaystyle=h_{\alpha\beta}+\widetilde{O}(r^{2})=\delta_{\alpha\beta}+\widetilde{O}(r^{2}),$

where we used Lemma 3.4. The definition of $\eta_{\alpha}$ requires $\langle\eta_{\alpha},dx_{j}\rangle=0$, which implies that

(3.78)

$$p_{i\alpha}g^{ij}+g^{j\alpha}=0.$$

Let $(\hat{g}_{ij})$ be the inverse of the matrix $(g^{ij})$ with $1\leq i,j\leq m-3$ such that $g^{ij}\hat{g}_{jk}=\delta_{ik}$. Multiplying by $\hat{g}_{jk}$, we have

(3.79)

$$p_{i\alpha}g^{ij}\hat{g}_{jk}+g^{j\alpha}\hat{g}_{jk}=0,$$

and hence

(3.80)

$$p_{k\alpha}=-g^{j\alpha}\hat{g}_{jk}.$$

We claim that for any $1\leq i,j\leq m-3$,

(3.81)

$$\hat{g}_{ij}-g_{ij}=\widetilde{O}(r^{2}).$$

In fact, since

(3.82)

$$g^{ij}g_{jk}+g^{i\alpha}g_{\alpha k}=\delta_{ik}.$$

Multipling by the inverse of the submatrix $(g^{ij})$,

(3.83)

$$\hat{g}_{li}(g^{ij}g_{jk}+g^{i\alpha}g_{\alpha k})=\hat{g}_{lk}.$$

So this implies that

(3.84)

$$\hat{g}_{lk}-g_{lk}=\hat{g}_{li}g^{i\alpha}g_{\alpha k}=\widetilde{O}(r^{2}).$$

Therefore, combining (3.75),(3.77), (3.80) and (3.84), we obtain

(3.85)		$\displaystyle p_{k\alpha}$	$\displaystyle=$	$\displaystyle-(g_{kj}+\widetilde{O}(r^{2}))g^{j\alpha}$
			$\displaystyle=$	$\displaystyle-g_{kj}g^{j\alpha}+\widetilde{O}(r^{3})$
			$\displaystyle=$	$\displaystyle h_{\alpha\beta}g_{k\beta}+\widetilde{O}(r^{3})$
			$\displaystyle=$	$\displaystyle g_{k\alpha}+\widetilde{O}(r^{3}).$

∎