Please consider the following hypersurface in $\mathbb{R}^{n+1}$:
$$-{c}^{2}{t}^{2}+\sum _{i=1}^{n}{x_{{i}}}^{2}=-{a}^{2} $$
Usin hyper-spherical coordinates the equation for the hypersurface is:
$$r^2+a^2=c^2t^2$$
where
$$ r^2 = \sum _{i=1}^{n}{x_{{i}}}^{2}$$
The corresponding metric is given by:
$$ ds^2 = {\frac {a^2}{r^2+a^2}}dr^2+ r^2 d\Omega_{n-1}$$
where $d\Omega_{n-1}$ is the metric of the hyper-sphere $S^{n-1}$
Making "experimental symbolic computation" with the package GrTensor, I am obtaining that
Ricci scalar:
$$ R = -{\frac {n(n-1)}{a^2}}$$
Kretschmann invariant :
$$ K = {\frac {2n(n-1)}{a^4}}$$
Einstein equation with cosmological constant:
$$ G_{{\mu,\nu}}+\Lambda\,g_{{\mu,\nu}}=0 $$
where
$$ \Lambda = -{\frac {(n-1)(n-2)}{2a^2}}$$
My question is: how to derive these equations starting from theorems of Differential Geometry?
$\newcommand\R{\mathbb{R}}$The curvature of the hypersurface depends on the Minkowski inner product you are using on $\mathbb{R}^{n+1}$. It appears to me that you are using $$ \langle ce_0,ce_0\rangle = -1,\ \langle e_0,e_i\rangle=0,\ \langle e_i,e_j\rangle=\delta_{ij},$$ where $1 \le i,j\le n$ and $(e_0,e_1, \cdots, e_n)$ is the standard basis of $\R^{n+1}$.
The standard hyperboloid, $$ -(ct)^2 + \langle x,x\rangle = -1, $$ has constant sectional curvature $-1$. Your hypersurface is the standard hyperboloid rescaled by the factor $a$. Therefore, it has constant sectional curvature equal to $-a^{-2}$.
It follows that if the metric tensor is $g_{\mu\nu}$ and the inverse metric tensor is $g^{\mu\nu}$, then the Riemann curvature tensor of this hypersurface is $$ R_{\mu\alpha\nu\beta} = -a^{-2}(g_{\mu\nu}g_{\alpha\beta} - g_{\mu\beta}g_{\nu\alpha}) $$ and Ricci curvature tensor is $$ R_{\mu\nu} = g^{\alpha\beta}R_{\mu\alpha\nu\beta} = -a^{-2}(g_{\mu\nu}g^{\alpha\beta}g_{\alpha\beta} - g_{\mu\beta}g^{\alpha\beta}g_{\nu\alpha}) = -(n-1)a^{-2}g_{\mu\nu}. $$ The scalar curvature is therefore $$ R = g^{\mu\nu}R_{\mu\nu} = -n(n-1)a^{-2}. $$ The Kretschmann invariant is defined to be $$ K = g^{\mu\rho}g^{\alpha\sigma}g^{\nu\tau}g^{\beta\xi}R_{\mu\alpha\nu\beta}R_{\rho\sigma\tau\xi} $$ The Einstein tensor is defined to be $$ R_{\mu\nu} - \frac{1}{2}Rg_{\mu\nu}. $$ These can be calculated by substituting in the formulas for the Riemann and Ricci curvature tensors.