Vector fields on $RP^m{\times}RP^n$

The span of a manifold is its maximum number of linearly independent vector fields. We discuss the question, still unresolved, of whether $\operatorname {span}(P^m\times P^n)$ always equals $\operatorname {span}(P^m)+\operatorname {span}(P^n)$. Here $P^n$ denotes real projective space. We use $BP$-cohomology to obtain new upper bounds for $\operatorname {span}(P^m\times P^n)$, much stronger than previously known bounds.