Following exercise:
Let $g_n, g_m$ be standard metrics on the spheres $S^n, S^m$, respectively. Now define the metric $g=\dfrac{1}{a}g_n+\dfrac{1}{b}g_m$ on $M:=S^n \times S^m$. I need to show, that it holds $K(\Pi)\in [0, max(a,b)]$ for the sectional curvature.
Now I already know that $K(\Pi)=0$ for $\Pi=span(x_n,x_m)$ with $x_n \in S^n, x_m \in S^m$.
My problem is, that I don't know what the length of a tangent vector in the standard metric is.. Can somebody give a hint?