In the answer https://math.stackexchange.com/q/4748125 of V. Semeria,

Taking $$(y_1,\dots,y_{n+1})=(x_1,\dots,x_n,\sum_{i=1}^{n+1}x_i^2 -R^2)$$ Write $(\vec{e}_1,\dots,\vec{e}_{n+1})$ the canonical basis of $\mathbb{R}^{n+1}$. For all $i\leq n$, it is easy to obtain $$ \partial y_i = \frac{\partial}{\partial y_i} = \vec{e}_i - \frac{x_i}{x_{n+1}}\vec{e}_{n+1} $$ Differentiate the $\partial y_i$ in euclidean $\mathbb{R}^{n+1}$, $$ \tilde{\nabla}_{\partial y_i}\partial y_j = \left(-\frac{x_ix_j}{x_{n+1}^3}-\frac{\delta_{ij}}{x_{n+1}}\right) \vec{e}_{n+1} .$$

He obtain $\tilde{\nabla}_{\partial y_i}\partial y_j = \left(-\frac{x_ix_j}{x_{n+1}^3}-\frac{\delta_{ij}}{x_{n+1}}\right) \vec{e}_{n+1}$ to compute $(\tilde{\nabla}_{\partial y_i}\partial y_j)^\top$($\top$ is the projection onto the tangent space of the sphere), we just need to compute $(\vec{e}_{n+1})^\top$, it is \begin{align*} &\sum^n_{i=1}\frac{\langle \vec{e}_{n+1},\partial y_i\rangle}{|\vec{e}_{n+1}||\partial y_i|} |\vec{e}_{n+1}|\frac{\partial y_i}{|\partial y_i|}=\sum^n_{i=1}\frac{\langle \vec{e}_{n+1},\partial y_i\rangle}{|\partial y_i|^2} {\partial y_i}\\ &=\sum^n_{i=1}\frac{\langle \vec{e}_{n+1},\vec{e}_i - \frac{x_i}{x_{n+1}}\vec{e}_{n+1}\rangle}{|\vec{e}_i - \frac{x_i}{x_{n+1}}\vec{e}_{n+1}|^2} {\partial y_i}=\sum^n_{i=1} \frac{-\frac{x_i}{x_{n+1}}}{1^2+(\frac{x_i}{x_{n+1}})^2}\partial y_i=\sum^n_{i=1} -\frac{x_i x_{n+1}}{x_i^2+x^2_{n+1}}\partial y_i. \end{align*}

Then, we obtain $\nabla_{\partial y_i}\partial y_j = \left(\frac{x_ix_j}{x_{n+1}^2}+\delta_{ij}\right)\sum_{k=1}^n\frac{x_k}{(x^2_k+x^2_{n+1})}\partial y_k, \quad\quad \text{i.e.} \quad\quad \Gamma_{ij}^k = \left(\frac{x_ix_j}{x_{n+1}^2}+\delta_{ij}\right)\frac{x_k}{(x^2_k+x^2_{n+1})}$.

But it is not consistent with $\Gamma_{ij}^k = \frac{1}{2}g^{kl}\left(\frac{\partial g_{il}}{\partial y_j} + \frac{\partial g_{jl}}{\partial y_i} - \frac{\partial g_{ij}}{\partial y_l}\right)= \left(\frac{x_ix_j}{x_{n+1}^2}+\delta_{ij}\right)\frac{x_k}{R^2}$ where $g_{ij} = \delta_{ij} + \frac{x_ix_j}{x_{n+1}^2}$ and $g^{ij} = \delta_{ij} - \frac{x_ix_j}{R^2}$.

These two methods should yield the same $\Gamma^k_{ij}$, where did I go wrong?

In his answer V. Semeria, starts by taking $$(y_1,\dots,y_{n+1})=\left(x_1,\dots,x_n,\sum_{i=1}^{n+1}x_i^2 -R^2\right)$$ Write $(\vec{e}_1,\dots,\vec{e}_{n+1})$ the canonical basis of $\mathbb{R}^{n+1}$. For all $i\leq n$, it is easy to obtain $$ \partial y_i = \frac{\partial}{\partial y_i} = \vec{e}_i - \frac{x_i}{x_{n+1}}\vec{e}_{n+1} $$ Differentiate the $\partial y_i$ in euclidean $\mathbb{R}^{n+1}$, $$ \tilde{\nabla}_{\partial y_i}\partial y_j = \left(-\frac{x_ix_j}{x_{n+1}^3}-\frac{\delta_{ij}}{x_{n+1}}\right) \vec{e}_{n+1} .$$ To compute $(\tilde{\nabla}_{\partial y_i}\partial y_j)^\top$ where $\top$ is the projection onto the tangent space of the sphere, He obtain $$ \tilde{\nabla}_{\partial y_i}\partial y_j = \left(-\frac{x_ix_j}{x_{n+1}^3}-\frac{\delta_{ij}}{x_{n+1}}\right) \vec{e}_{n+1}, $$ and after we just need to compute $(\vec{e}_{n+1})^\top$, that is $$ \begin{split} \sum^n_{i=1}\frac{\langle \vec{e}_{n+1},\partial y_i\rangle}{|\vec{e}_{n+1}||\partial y_i|} |\vec{e}_{n+1}|\frac{\partial y_i}{|\partial y_i|} & =\sum^n_{i=1}\frac{\langle \vec{e}_{n+1},\partial y_i\rangle}{|\partial y_i|^2} {\partial y_i}\\ &=\sum^n_{i=1}\frac{\langle \vec{e}_{n+1},\vec{e}_i - \frac{x_i}{x_{n+1}}\vec{e}_{n+1}\rangle}{|\vec{e}_i - \frac{x_i}{x_{n+1}}\vec{e}_{n+1}|^2} {\partial y_i}\\ &=\sum^n_{i=1} \frac{-\frac{x_i}{x_{n+1}}}{1^2+(\frac{x_i}{x_{n+1}})^2}\partial y_i\\ & =\sum^n_{i=1} -\frac{x_i x_{n+1}}{x_i^2+x^2_{n+1}}\partial y_i. \end{split} $$ Then, we obtain $$ \nabla_{\partial y_i}\partial y_j = \left(\frac{x_ix_j}{x_{n+1}^2}+\delta_{ij}\right)\sum_{k=1}^n\frac{x_k}{(x^2_k+x^2_{n+1})}\partial y_k, $$ i.e. $$ \Gamma_{ij}^k = \left(\frac{x_ix_j}{x_{n+1}^2}+\delta_{ij}\right)\frac{x_k}{(x^2_k+x^2_{n+1})}. $$ But this result is not consistent with the following: $$ \Gamma_{ij}^k = \frac{1}{2}g^{kl}\left(\frac{\partial g_{il}}{\partial y_j} + \frac{\partial g_{jl}}{\partial y_i} - \frac{\partial g_{ij}}{\partial y_l}\right)= \left(\frac{x_ix_j}{x_{n+1}^2}+\delta_{ij}\right)\frac{x_k}{R^2} $$ where

• $g_{ij} = \delta_{ij} + \frac{x_ix_j}{x_{n+1}^2}$ and
• $g^{ij} = \delta_{ij} - \frac{x_ix_j}{R^2}$.

These two methods should yield the same $\Gamma^k_{ij}$, where did I go wrong?