Let $M$ and $N$ be two Riemannian manifolds with Riemannian metrics $g$, $h$ respectively. We consider the product $M \times N$ with metric $g \oplus h$. By the metric we get an isomorphism of bundles $T(M \times N) = TM \times TN$.
My question is: is this also true for the cotangent bundle?
I.e. is $T^{*}(M \times N) = T^{*}M \times T^{*}N$?
I hope for a lot of answers and want to thank in advance.