We consider a hyperbolic half-plane (also called Poincar ́e upper half plane),
$$\mathbb{H}^{2}=\left\{ \left(x,y\right)\Biggl|y>0\right\}$$ .
It has got a simple 2D Riemannian metric with open domain, given by:
$$ds^{2}=\frac{1}{\sigma^{2}y^{2}}dx^{2}+\frac{1}{\sigma^{2}y^{2}}dy^{2}$$.
We can look at a theoram called Taylor series theoram, [http://web.math.ucsb.edu/~moore/riemanniangeometry2011.pdf||page 71] with the help of which one can expand Riemannian metric, $$g_{ij}$$ in terms of normal coordinates at a point p on the Manifold. This can be expressed as:
$$g_{ij}=\delta_{ij}-\frac{1}{3}\sum_{k,l=1}^{n}R_{ikjl}\left(p\right)x^{k}x^{l}+\left(higher-order-terms\right)$$.
The above equation can be written alternatively as:
$$g_{ij}=\delta_{ij}+\frac{1}{2}\sum_{k,l=1}^{n}\frac{\partial^{2}g_{ij}}{\partial x^{k}\partial x^{l}}\left(p\right)x^{k}x^{l}+\left(higher-order-terms\right)$$
We have the components of the metric tensor in hyperbolic coordinates is given by:
$$g_{xx}=g_{yy}=\frac{1}{\sigma^{2}y^{2}}.$$
This implies the components of the metric tensor in the normal coordinates will be:
$$g_{xx} =\delta_{xx}+\frac{1}{2}\sum_{k,l=1}^{2}\frac{\partial^{2}g_{xx}}{\partial x^{k}\partial x^{l}}\left(p\right)x^{k}x^{l} =\delta_{xx}+\frac{1}{2}\left[\frac{\partial^{2}g_{xx}}{\partial x^{1}\partial x^{1}}\left(p\right)x^{1}x^{1}+\frac{\partial^{2}g_{xx}}{\partial x^{1}\partial x^{2}}\left(p\right)x^{1}x^{2}+\frac{\partial^{2}g_{xx}}{\partial x^{2}\partial x^{1}}\left(p\right)x^{2}x^{1}+\frac{\partial^{2}g_{xx}}{\partial x^{2}\partial x^{2}}\left(p\right)x^{2}x^{2}\right] =1+\frac{1}{2}\left[\frac{\partial^{2}g_{xx}}{\partial x\partial x}\left(p\right)x^{2}+\frac{\partial^{2}g_{xx}}{\partial x\partial y}\left(p\right)xy+\frac{\partial^{2}g_{xx}}{\partial y\partial x}\left(p\right)yx+\frac{\partial^{2}g_{xx}}{\partial y\partial y}\left(p\right)y^{2}\right] =1+\frac{1}{2}\left[\frac{\partial^{2}g_{xx}}{\partial y\partial y}\left(p\right)y^{2}\right] =1-2\frac{1}{2}\left[\frac{1}{\sigma^{2}}y^{-3}\left(p\right)y^{2}\right] =1-\left[\frac{1}{\sigma^{2}}y^{-3}\left(p\right)y^{2}\right] g_{xx} =1-\left[\frac{1}{\sigma^{2}y^{3}}\left(p\right)y^{2}\right]$$
It is understood that:
$$g_{yy}=1-\left[\frac{1}{\sigma^{2}y^{3}}\left(p\right)y^{2}\right]$$
Here we are taking the point p, $$x\left(0\right)=y\left(0\right)=1$$.
This implies,
$$g_{xx} =g_{yy}=1-\left[\frac{1}{\sigma^{2}y^{3}}\left(1,1\right)y^{2}\right] =1-\left[\frac{1}{\sigma^{2}}y^{2}\right]$$
Am I doing the right way?