Riemannian Geometry Part 3

Problem 6.74 Find the Riemann-Christoffel curvature tensor of the Riemannian manifold $(U, g)$, where $U$ denotes the unit open disk of the plane $\mathbb{R}^{2}$ and $$ g=\frac{1}{1-x^{2}-y^{2}}\left(\mathrm{~d} x^{2}+\mathrm{d} y^{2}\right) $$

Problem 6.75 Consider on $\mathbb{R}^{3}$ the metric $$ g=\mathrm{e}^{2 z}\left(\mathrm{~d} x^{2}+\mathrm{d} y^{2}+\mathrm{d} z^{2}\right) $$ Compute $R\left(\frac{\partial}{\partial x}, \frac{\partial}{\partial z}, \frac{\partial}{\partial x}, \frac{\partial}{\partial z}\right)$, where $R$ denotes the Riemann-Christoffel curvature tensor.

Problem 6.76 Let $(M, g)$ be a Riemannian $n$-manifold. Consider an orthonormal basis $\left\{e_{1}, \ldots, e_{n-1}, X\right\}$ of $T_{p} M, p \in M$. Let $P_{i}$ be the plane section generated by $e_{i}$ and $X ; K\left(P_{i}\right)$ the sectional curvature of $P_{i} ;$ and $\mathbf{r}$ the Ricci tensor. Prove that $$ \mathbf{r}(X, X)=\sum_{i=1}^{n-1} K\left(P_{i}\right) $$