Recall the following definitions. The smooth manifold $\mathbb{S}_R^n$ is defined as a hypersurface in $\mathbb{R}^{n+1}$, and if $\iota:\mathbb{S}_R^n\to\mathbb{R}^{n+1}$ is the inclusion map and $\bar{g}$ the Riemmanian metric on $\mathbb{R}^{n+1}$, then we let $\mathring{g}_R:=\iota^*\bar{g}$ be the Riemmanian metric on $\mathbb{S}_R^n$, so that $(\mathbb{S}_R^n,\mathring{g}_R)$ is a Riemannian manifold.
We also define the stereographic projections $$\varphi_\pm:\mathbb{S}_R^n\setminus\{\mp R e_{n+1}\}\to\mathbb{R}^n,\ (x^1,\ldots,x^{n+1})\mapsto\left(\frac{Rx^1}{R\pm x^{n+1}},\ldots,\frac{Rx^n}{R\pm x^{n+1}}\right)$$ which have inverses $$\varphi_\pm^{-1}:\mathbb{R}^n\to\mathbb{S}_R^n\setminus\{\mp R e_{n+1}\},\ (y^1,\ldots,y^n)\mapsto\left(\frac{2R^2 y^1}{R^2+{\vert y\vert}^2},\ldots,\frac{2R^2 y^n}{R^2+{\vert y\vert}^2},\pm\frac{R^3-R{\vert y\vert}^2}{R^2+{\vert y\vert}^2}\right)$$ which allows us to define coordinates $(y_\pm^1,\ldots,y_\pm^n)$ on $\mathbb{S}_R^n\setminus\{\mp R e_{n+1}\}$, though we will typically write $y^j$ in place of $y_\pm^j$ for convenience. If we let $g=\mathring{g}_R$, then a direct computation tells us that $$g_{ij} = \frac{4R^4\delta_{ij}}{(R^2+{\vert y\vert}^2)^2}$$ in coordinates. The formulae for the Chritsoffel symbols are only slightly more complicated. In coordinates, we have $$\Gamma_{ij}^k = -\frac{2y^i}{R^2+{\vert y\vert}^2}\ \mathrm{if}\ j=k,\ \Gamma_{ij}^k = -\frac{2y^j}{R^2+{\vert y\vert}^2}\ \mathrm{if}\ i=k$$ while $$\Gamma_{ij}^k = \frac{2y^k}{R^2+{\vert y\vert}^2}\ \mathrm{if} i=j\ne k$$ and $\Gamma_{ij}^k=0$ otherwise. Now, we can express the Riemannian curvature tensor $\mathrm{R}$ in coordinates as $$\mathrm{R}_{ijk}^m = \Gamma_{ik}^l\Gamma_{jl}^m - \Gamma_{jk}^l\Gamma_{il}^m + \partial_j\Gamma_{ik}^m - \partial_i\Gamma_{jk}^m.$$
Unlike with previous computations, however, I see of no obvious simplification that can be made to this expression. So this is my question: is there a simple formula for $\mathrm{R}$ that works for all $n$? Is there some simplification that can be made to reduce the possible forms of $\mathrm{R}_{ijk}^m$ to a small number of cases? Or do we simply have increasingly convoluted and messy expressions for $R$ as $n\to\infty$?
For that matter, is there a simpler way of expressing $\mathrm{Ric}_{ij}=\mathrm{R}_{kij}^k$, or $K = g^{ij}\mathrm{Ric}_{ij} = g^{ik}\mathrm{R}_{ijk}^j$?
It may possibly be useful to note that $$ \sqrt{\det(g_{ij})}=\left(\frac{2R^2}{R^2+\lvert y\rvert^2}\right)^n $$
It may also possibly be useful to note that
$$ \Delta f = -\left(\frac{R^2+\lvert y\rvert^2}{2R^2}\right)^2\sum_{j=1}^n\left(\frac{\partial^2f}{(\partial y^j)^2} - \frac{2(n-2)y^j}{R^2+\lvert y\rvert^2}\frac{\partial f}{\partial y^j}\right) $$ where $\Delta$ is the Laplace-Beltrami operator.
