In Brendle's paper (S. Brendle, Convergence of the Yamabe flow in dimension 6 and higher, Invent. Math. 170 (3) (2007), 541-576.), let $p \in M$, he defined the following set $$\mathcal{Z}=\Big\{x \in M; \limsup_{x\to p}d_g(p,x)^{2-d}|W_g(x)|=0\Big\},$$ where $d=[\frac{n-2}{2}]$ for any $n\geq 6$. In his proof of Proposition 19 (on pages $568-570$), he used the following fact: choose conformal normal coordinates around $p \in \mathcal{Z}$, up to a conformal factor of $g$, near $p$ there hold $$\det g(x)=1+O(|x|^{2d+2})$$ and $$ g_{ik}=\delta_{ik}+O(|x|^{d+1}).\qquad\qquad \qquad (1) $$ In general, it is easy to show that $(1)$ and the properties of conformal normal coordinates can imply $p \in \mathcal{Z}$. But its inverse is not easy. So, can you give me some clues to show the above assertion? I guess we perhaps should use the decomposition of Riemannian curvature tensor, the Taylor's expansion of the metric near $p$ or something else. I tried it and can not find a rigorous proof of it.
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