What is the metric on the $n$-sphere in stereographic projection coordinates?

The metric on the $n$-sphere is the metric induced from the ambient Euclidean metric. Find the metric, $d\Omega^2_n$, on the $n$-sphere and the volume form, $\Omega_{S_n}$ , of $S^n$ in terms of the stereographic coordinates on $U_N =S^n − (0, . . . , 0, 1)$.

The stereographic projection coorinates $u_j$ are given by $u_j=\frac{x_j}{1-x_{n+1}}$ for $j=1,\dots,n$.

We also have $x_j=\frac{2u_j}{1+\sum_{k=1}^nu_k^2}$ and $x_{n+1}=\frac{1-\sum_{k=1}^nu_k^2}{1+\sum_{k=1}^nu_k^2}$

I took differentials and put them into the expression for the Euclidean metric put things are getting messy so I'm not sure if it's right. I also have to find the components of the Levi-Civitta connection, curvature tensor, Ricci tensor.

Solution 1:

Let's find the metric on the sphere in stereographic projection. We are going to give the chart and its inverse for $S^n(R)$, the sphere centre at the origin and radius $R$. The chart $\varphi:S^n(R)\subset\mathbb{R}^{n+1} \to \mathbb{R}^n$ is given by \begin{equation} \varphi(\xi,\tau)=\left(\frac{R\xi}{R-\tau}\right), \end{equation} where $\xi=(\xi_1,\ldots,\xi_n)\in\mathbb{R}^n, \tau\in \mathbb{R}$, $|\cdot|$ is the euiclidean norm, and $\xi, \tau$ are such that $|\xi|^2 + |\tau|^2=R^2$.

Its inverse map is the parametrization $\varphi^{-1}:\mathbb{R}^n\to S^n(R)$ given by \begin{equation} \varphi^{-1}(u)=\left(\frac{2R^2u}{|u|^2+R^2},R\frac{|u|^2-R^2}{|u|^2+R^2}\right), \end{equation} with $u=(u_1,\ldots,u_n)\in\mathbb{R}^n$.

Let $V\in T_u\mathbb{R}^n$ a tangent vector at $u$. Then it can be written as \begin{equation} V = \sum_{k=1}^{n}V(u_k)\left.\frac{\partial}{\partial u_k}\right|_u \end{equation}

Now, since $d_u\varphi^{-1}:T_u\mathbb{R}^n\to S^n(R)$ we compute: \begin{equation} d_u\varphi^{-1}(V)= \sum_{k=1}^{n}V(u_k) d_u\varphi^{-1}\left(\left.\frac{\partial}{\partial u_k}\right|_u\right), \end{equation} since $d_u\varphi(\partial/\partial u_k|_u)\in T_pS^n\subset\mathbb{R}^{n+1}$ then, is linear combination of the $\{\partial/\partial \xi_j,\partial/\partial\tau\}$, we omit the $u$ subscript, we have: \begin{align*} d_u\varphi^{-1}\left( \frac{\partial}{\partial u_k} \right)&= \sum_{j=1}^n d_u\varphi^{-1}\left(\frac{\partial}{\partial u_k}\right)(\xi_j)\frac{\partial}{\partial \xi_j} + d_u\varphi^{-1}\left(\frac{\partial}{\partial u_k}\right)(\tau)\frac{\partial}{\partial\tau}\\ &= \sum_{j=1}^n \frac{\partial}{\partial u_k}(\xi_j\circ\varphi^{-1})\frac{\partial}{\partial \xi_j} + \frac{\partial}{\partial u_k}(\tau\circ\varphi^{-1})\frac{\partial}{\partial\tau},\\ \end{align*} now, from the above formula we have \begin{align*} d_u\varphi^{-1}(V)&= \sum_{k=1}^{n}V(u_k) \sum_{j=1}^n \frac{\partial}{\partial u_k}(\xi_j\circ\varphi^{-1})\frac{\partial}{\partial \xi_j} +\sum_{k=1}^{n}V(u_k) \frac{\partial}{\partial u_k}(\tau\circ\varphi^{-1})\frac{\partial}{\partial\tau}\\ &= \sum_{j=1}^{n}\sum_{k=1}^n V(u_k)\frac{\partial}{\partial u_k}(\xi_j\circ\varphi^{-1})\frac{\partial}{\partial \xi_j} +\sum_{k=1}^{n}V(u_k) \frac{\partial}{\partial u_k}(\tau\circ\varphi^{-1})\frac{\partial}{\partial\tau}\\ &= \sum_{j=1}^{n}V(\xi_j\circ\varphi^{-1})\frac{\partial}{\partial \xi_j} +V (\tau\circ\varphi^{-1})\frac{\partial}{\partial\tau}\\ \end{align*}

We are going to abuse on the notation and we'll write $V_k$ instead of $V(u_k)$ and also we write $V(\xi_j)$ instead of $V(\xi_j\circ \varphi^{-1})$. Then the tensor metric is $d\Omega_n^2 = \sum_{j=1}^n V(\xi_j)^2 + V(\tau)^2$. Then \begin{align*} V(\xi_j)&=\sum_{k=1}^n V_k\frac{\partial }{\partial u_k}\left(\frac{2R^2u_j}{|u|^2+R^2}\right)\\ &=\sum_{k=1}^n V_k\left(\frac{2R^2\delta_{jk}}{|u|^2+R^2}- \frac{4R^2u_ju_k}{(|u|^2+R^2)^2}\right)\\ &=\frac{2R^2V_j}{|u|^2+R^2}- \frac{4R^2u_j}{(|u|^2+R^2)^2}\sum_{k=1}^n V_k u_k \\ &=\frac{2R^2V_j}{|u|^2+R^2}- \frac{4R^2u_j}{(|u|^2+R^2)^2}\langle V, u\rangle \\ \end{align*} where we used that $\delta_{jk}$ equals $1$ if $j=k$ and vanished otherwise. In the same way we have \begin{align*} V(\tau)&=\sum_{k=1}^n V_k\frac{\partial }{\partial u_k}\left(R\frac{|u|^2-R^2}{|u|^2+R^2}\right)\\ &=\sum_{k=1}^n V_k\left(\frac{2Ru_{k}}{|u|^2+R^2}- \frac{2R(|u|^2-R^2)u_k}{(|u|^2+R^2)^2}\right)\\ &=\frac{2R}{|u|^2+R^2}\langle V,u\rangle- \frac{2R(|u|^2-R^2)}{(|u|^2+R^2)^2}\langle V,u\rangle\\ &=\frac{2R(|u|^2+R^2)-2R(|u|^2-R^2)}{(|u|^2+R^2)^2}\langle V,u\rangle\\ &=\frac{4R^3}{(|u|^2+R^2)^2}\langle V,u\rangle\\ \end{align*}

Now we are able to compute the metric: \begin{align*} d\Omega_n^2 &= \sum_{j=1}^n V(\xi_j)^2 + V(\tau)^2\\ &= \sum_{j=1}^n \left(\frac{2R^2V_j}{|u|^2+R^2}- \frac{4R^2u_j}{(|u|^2+R^2)^2}\langle V, u\rangle\right)^2+ \left(\frac{4R^3}{(|u|^2+R^2)^2}\langle V,u\rangle\right)^2\\ &=\sum_{j=1}^n\left\{\frac{4R^4V_j^2}{(|u|^2+R^2)^2}- \frac{16R^4V_ju_j}{(|u|^2+R^2)^3}\langle V,u\rangle+ \frac{16R^4u_j^2}{(|u|^2+R^2)^4}\langle V,u\rangle^2\right\}+ \frac{16R^6}{(|u|^2+R^2)^4}\langle V,u\rangle^2\\ &=\frac{4R^4|V|^2}{(|u|^2+R^2)^2}- \frac{16R^4}{(|u|^2+R^2)^3}\langle V,u\rangle^2+ \frac{16R^4|u|^2}{(|u|^2+R^2)^4}\langle V,u\rangle^2+ \frac{16R^6}{(|u|^2+R^2)^4}\langle V,u\rangle^2\\ &=\frac{4R^4|V|^2}{(|u|^2+R^2)^2} \end{align*}

The we conclude that in stereographic projection \begin{equation} d\Omega_n^2 (d\varphi^{-1}(V))= \frac{4R^4}{(|u|^2+R^2)^2}ds^2(V) \end{equation} where $ds^2$ stands for the stantdard metric in $\mathbb{R}^n$.

This calculation can be found in John M. Lee's Riemannian Manifolds. An introduction to Curvature Chap. 3.