# Curvature of Hyperbolic Space

I'm trying to prove that hyperbolic space has constant sectional curvature $-1$, but keep running into difficulties. Could someone show me a way out?

I've been given the metric

$$g_{ij}=\frac{4\delta_{ij}}{(1-\sum_{i=1}^{n} (x^i)^2)^2}$$

I know that wlog I need to calculate the component of the Riemann tensor $R^i_{jij}$ (no summation) and show that this is $-\frac{1}{4}$ at the origin.

So I expand the metric to second order at $x=0$ getting

$g_{ij}=4\delta_{ij}-8x^ix^j$. Now from work I did earlier for calculating the sectional curvature of the sphere I know that this will give me $R^i_{jij}=-2$.

But this value is too large in modulus by a factor of $8$! What on earth am I doing wrong here? I genuinely can't see how to do this correctly.

Note: please don't reply with a good alternative solution to the problem, not involving the calculation of the Riemann tensor. I'm aware of these, but want to solve this problem as stated for my own satisfaction!

Given the formula for the sectional curvature, we can make the tangent vectors orthonormal by taking $u= \frac{x^i}{4\delta_{ij}- 8x^i 8x^j}, v= \frac{x^j}{4\delta_{ij}- 8x^i 8x^j}$. Now we can simply use the standard Riemann curvature $K(u,v)= \left \langle R(Du,Dv)Dv,Du \right \rangle$, since the tangent vectors are along the tangent space. This definition still involves the Riemann curvature tensor as required in your question. – Jaivir Baweja Jan 21 '13 at 18:21