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Lastly we have the hyperbolic space $D^n=\{z\in\mathbb{C}^n:|z|<1\}$. We endowe it with the Hermitian metric coming from the following global Kähler potential

$$\mathcal{K}=-\log(1-|z|^2)$$

Repeating the exact same steps done for the projective space, we get

$$h_{jk}=-2\dfrac{\partial ^2}{\partial z_j\partial \bar{z}_k}\log(1-|z|^2)=-2\dfrac{\partial }{\partial z_j}\left(\dfrac{-z_k}{1-|z|^2}\right)=2\dfrac{\delta_{jk}(1-|z|^2)+\bar{z}_j z_k}{(1-|z|^2)^2}$$

$$\tilde{h}=-2\sum_{jk}\dfrac{\partial ^2}{\partial z_j\partial \bar{z}_k}\log(1-|z|^2)\mathrm{d} z_j\otimes\mathrm{d} \bar{z}_k$$

$$g=-\sum_{jk}\left(\dfrac{\partial ^2}{\partial z_j\partial \bar{z}_k}\log(1-|z|^2)\mathrm{d} z_j\otimes\mathrm{d} \bar{z}_k+\dfrac{\partial ^2}{\partial \bar{z}_j\partial z_k}\log(1-|z|^2)\mathrm{d} \bar{z}_j\otimes\mathrm{d} z_k\right)$$

$$\omega=-\mathrm{i}\sum_{jk}\left(\dfrac{\partial ^2}{\partial z_j\partial \bar{z}_k}\log(1-|z|^2)\mathrm{d} z_j\otimes\mathrm{d} \bar{z}_k-\dfrac{\partial ^2}{\partial \bar{z}_j\partial z_k}\log(1-|z|^2)\mathrm{d} \bar{z}_j\otimes\mathrm{d} z_k\right)=-\mathrm{i}\partial\bar{\partial}\log(1-|z|^2)$$

$$\det g=\dfrac{1}{(1-|z|^2)^{n+1}},\qquad\rho=-(n+1)\omega,\qquad R_{j\bar{k}}=-(n+1)g_{j\bar{k}},\qquad Sc=-n(n+1)$$

and this is again an Einstein metric, of constant $-(n+1)$.