Let $\varphi:\mathbb R^n\longrightarrow \mathbb S^n$ the inverse of the stereographic projection, i.e. $$\varphi(y)=\left(\frac{2y}{\|y\|^2+1},\frac{\|y\|^2-1}{\|y\|^2+1}\right).$$
What I'm trying to compute is $g:= \varphi^* g_{sphere}$. My course gave me $$g_y(u,u)=(g_{sphere})_{\varphi(y)}(\mathrm d _y\varphi\cdot u,\mathrm d _y \varphi\cdot u),$$ but I don't understand this formula.
So is $\mathrm d _y\varphi\cdot u$ a scalar product ? Wouldn't it be $$g_y(u,v)=(g_{sphere})_{\varphi(y)}(\mathrm d _y\varphi(u),\mathrm d _y \varphi(v))\ ?$$ and Finally, how can I compute it ? I don't really see how it works... well, to me $$\mathrm d _y\varphi=\sum_{i=1}^n\frac{\partial \varphi}{\partial y^i}\mathrm d y_i$$ but it looks strange here. Any help would be appreciate.
The formula "$d _y\varphi\cdot u$" is the application of the operator $d_y \varphi : T_y \mathbb{R}^n \to T_{\varphi(y)} \mathbb{S}^n$ to the tangent vector $u \in T_y \mathbb{R}^n$. It is indeed another way of writing $(d_y\varphi)(y)$, only with less parentheses. It is a common notation in differential geometry, where the operators can have somewhat big names and the result can be a bit ambiguous; when you write $d_y \varphi (u)$, is it $d_y \varphi$ applied to $u$, or $d_y$ applied to $\varphi(u)$? With the context it's obvious the correct answer is the first, but it's just to be on the safe side.
As to how to compute it, the expression you wrote is not optimal for this. Indeed $\frac{2y}{\|y\|^2 + 1}$ is a vector of $\mathbb{R}^n$, whereas $\frac{\|y\|^2-1}{\|y\|^2+1}$ is a number, and you implicitly use the identification $\mathbb{R}^n \times \mathbb{R} = \mathbb{R}^{n+1}$ (with $\mathbb{S}^n \subset \mathbb{R}^{n+1}$).
I'll explain for $n=2$, you do the rest. For $n=2$, you $\varphi$ becomes: $$\varphi(y_1,y_2) = \left( \frac{2 y_1}{y_1^2+y_2^2+1}, \frac{2 y_2}{y_1^2+y_2^2+1}, \frac{y_1^2+y_2^2-1}{y_1^2+y_2^2+1} \right) = (\varphi_1(y), \varphi_2(y), \varphi_3(y)).$$
Let $y = (y_1, y_2) \in \mathbb{R}^2$, then $d_y \varphi \cdot u$ is a linear operator acting on $u = (u_1, u_2) \in T_y \mathbb{R}^n$ with matrix: $$\begin{pmatrix} \frac{\partial \varphi_1}{\partial y_1}(y) & \frac{\partial \varphi_1}{\partial y_2}(y) \\ \frac{\partial \varphi_2}{\partial y_1}(y) & \frac{\partial \varphi_2}{\partial y_2}(y) \\ \frac{\partial \varphi_3}{\partial y_1}(y) & \frac{\partial \varphi_3}{\partial y_2}(y) \\ \end{pmatrix}$$
In other words, $d_y \varphi \cdot u \in T_{\varphi(y)} \mathbb{S}^2 \subset T_{\varphi(y)} \mathbb{R}^3 \cong \mathbb{R}^3$ is: $$\left( \frac{\partial \varphi_1}{\partial y_1}(y) \cdot u_1 + \frac{\partial \varphi_1}{\partial y_2}(y) \cdot u_2, \frac{\partial \varphi_2}{\partial y_1}(y) \cdot u_1 + \frac{\partial \varphi_2}{\partial y_2}(y) \cdot u_2, \frac{\partial \varphi_3}{\partial y_1}(y) \cdot u_1 + \frac{\partial \varphi_3}{\partial y_2}(y) \cdot u_2 \right)$$ where now $\cdot$ is just multiplication of two real numbers.