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 manifolds 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.
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 $$h_{ij}(0)=g(\nabla_{\partial y^{i}}\partial y^{n},\partial_{y^{j}})=\Gamma_{in}^{k}g_{kj}=0.$$
This means its second fundamental form is 0.But this seems strange to me.
Is this true?Any help will be thanked.
