I'm reading Escobar's The Yamabe Problem On Manifolds With Boundary.
He says
Let $(y_{1},\cdots,y_{n})$ be normal coordinates around $0\in \partial M$, such that $\eta(0)=-\frac{\partial}{\partial y_{n}}$, and second fundamental form of $\partial M$ at 0 has a diagonal form.
Here $M$ is a Riemannian manifold with boundary and $0$ is a nonumbilic point on $\partial M$.$\eta$ represents the outward normal.
I wonder how can we take such a normal coordinates. As I know,normal coordinates can be taken at boundary point only when the boundary is locally totally geodesic. And in this case the second fundamental form must be 0.
In fact if we take a normal coordinates at $0$, which satisfies $g_{ij}(0)=\delta_{ij},\Gamma_{ij}^{k}(0)=0.$
Then we compute $$h_{ij}(0)=g(\nabla_{\partial y^{i}}\partial y^{n},\partial_{y^{j}})=\Gamma_{in}^{k}g_{kj}=0.$$
So how can we take such a normal coordinate? Any help will be thanked.