Let $M_1$ and $M_2$ be Riemannian manifolds and consider the cartesian product $M_1 \times M_2$ with the profuct structure .Let $\pi_1: M_1 \times M_2 \to M_1$ and $\pi_2: M_1 \times M_2 \to M_2$be the natural projections.Introduce on $\ M_1 \times M_2$ a Riemannian metric as follows:
$$\langle u,v \rangle_{(p,q)} = \langle {dπ_1}_* u,{dπ_1}_* v \rangle_p + \langle {dπ_2}_*u,{dπ_2}_* v \rangle_q$$ .
How can I verify that this is actually a Riemannian metric.
To show this first I need to show that this bilinear and smooth. Am I right?
If yes then
(1) since derivative map is linear and inner product is bilinear do this is linear.is it ok?
(2) but how can I show that this is smooth?
Need your help.Please explain in detail.I am totally new in the subject.