First, we prove Item (1). By (3.51) we have

(3.97)

$$(-1)^{m+1}*\dvol_{T}=\lambda\cdot\dvol_{N}$$

for a function $\lambda>0$. The function $\lambda$ is given by

(3.98)

$$\lambda=\frac{|\dvol_{T}|^{2}\sqrt{\det(g)}}{\sqrt{\det(g_{ij}^{P})}}=\sqrt{\det(g)}\det(g^{ij})\sqrt{\det(g_{ij}^{P})}.$$

Now we compute the expansion of $\lambda$. Applying the expansions of $g_{ij}$, $g_{\alpha\beta}$ and $g_{i\alpha}$ in Lemma 3.4, one can directly obtain the following,

(3.99)		$\displaystyle\det(g)$	$\displaystyle=\det(g_{\alpha\beta})\cdot\det(g_{ij})+\widetilde{O}(r^{2}),$
(3.100)		$\displaystyle\det(g_{\alpha\beta})$	$\displaystyle=1+\widetilde{O}(r^{2}),$
(3.101)		$\displaystyle\det(g_{ij})$	$\displaystyle=\det(g_{ij}^{P})\cdot(1+2H^{\alpha}y_{\alpha})+\widetilde{O}(r^{2}).$

Plugging (3.100) and (3.101) into (3.99),

(3.102)

$$\det(g)=\det(g_{ij}^{P})\cdot(1+2H^{\alpha}y_{\alpha})+\widetilde{O}(r^{2}).$$

Let $(h_{ij})$ be the inverse of the matrix $(g_{ij})$. Since $g^{ij}=h_{ij}+\widetilde{O}(r^{2})$ by (3.76), so it follows that

(3.103)

$$\det(g^{ij})=\det(h_{ij})+\widetilde{O}(r^{2})=(\det(g_{ij}))^{-1}+\widetilde{O}(r^{2}).$$

Plugging (3.101) into the above,

(3.104)

$$\det(g^{ij})=\det(g_{ij}^{P})^{-1}\cdot(1-2H^{\alpha}y_{\alpha})+\widetilde{O}(r^{2}).$$

Therefore, substituting (3.102) and (3.104) into (3.98),

(3.105)

$$\lambda=1-H^{\alpha}y_{\alpha}+\widetilde{O}(r^{2}),$$

which completes the proof of Item (1).

Now we prove Item (2). For each $\alpha\in\{1,2,3\}$, we can write

(3.106)

$$*(\eta_{\alpha}\wedge\dvol_{T})=\lambda_{1}\cdot\eta_{\widehat{\alpha}}+\lambda_{2}\cdot\eta_{\widehat{\alpha+1}}+\lambda_{3}\cdot\eta_{\widehat{\alpha+2}}.$$

Taking point-wise wedge product with $\eta_{\alpha}\wedge\dvol_{T}$, and noticing $\eta_{\widehat{\alpha+1}}\wedge\eta_{\alpha}$, $\eta_{\widehat{\alpha+2}}\wedge\eta_{\alpha}$ are both zero, then we obtain

(3.107)

$$\lambda_{1}\cdot\eta_{\widehat{\alpha}}\wedge\eta_{\alpha}\wedge\dvol_{T}=(\eta_{\alpha}\wedge\dvol_{T})\wedge*(\eta_{\alpha}\wedge\dvol_{T}).$$

(3.108)

$\displaystyle\lambda_{1}\cdot\dvol_{N}\wedge\dvol_{T}=|\eta_{\alpha}\wedge\dvol_{T}|^{2}\dvol_{g}$

Therefore, by (3.51) and (3.87) we get

(3.109)

$$\lambda_{1}=\frac{|\eta_{\alpha}\wedge\dvol_{T}|^{2}\sqrt{\det(g)}}{\sqrt{\det(g_{ij}^{P})}}=1-H^{\alpha}y_{\alpha}+\widetilde{O}(r^{2}),$$

Similarly taking wedge product with $\eta_{\alpha+1}\wedge\dvol_{T}$ and $\eta_{\alpha+2}\wedge\dvol_{T}$ respectively, and again by (3.87) we obtain that

(3.110)

$$\lambda_{2}=\widetilde{O}(r^{2}),\lambda_{3}=\widetilde{O}(r^{2}).$$

These imply that

(3.111)

$$*(\eta_{\alpha}\wedge\dvol_{T})=\lambda_{1}\cdot\eta_{\widehat{\alpha}}+\lambda_{2}\cdot\eta_{\widehat{\alpha+1}}+\lambda_{3}\cdot\eta_{\widehat{\alpha+2}},$$

where $\lambda_{1}=1+H^{\alpha}y_{\alpha}+\widetilde{O}(r^{2})$, $\lambda_{2}=\widetilde{O}(r^{2})$ and $\lambda_{3}=\widetilde{O}(r^{2})$. ∎