Suppose $\pi:M\times N\to M$ defined by $\pi(x,y)=x\ ,\forall (x,y)\in M\times N$ where $M,N$ are smooth manifolds of dimension m and n respectively. To prove that differential of $\pi$ is itself.
My try: In local coordinates Jacobian of $\pi$ is $\begin{bmatrix} I_m & 0\end{bmatrix}_{m\times (m+n)}$. From here how can I conclude?