Vector fields on \(\mathbb RP^m\times \mathbb RP^n\) (Q2846862)

Let \(P^n\) be the real projective space of dimension \(n\) and let \(\nu(n)\) be the exponent of \(2\) in \(n\). Observe that if \(\nu(n+1) = 4a +b, \, 0 \leq b \leq 3\), then \(V(n) = 8a +2^b -1\).NEWLINENEWLINEThis work considers the span\((P^m \times P^n)\), where the span of a manifold means the maximal number of linearly independent vector fields on it. The author proves two results. The first one states that span\((P^m \times P^n) = V(m) + V(n)\), if neither \(m+1\) or \(n+1\) is divisible by \(16\) or if \( m =1,3\) or \(7\). The second result, considered the main result of this paper by the author, gives upper bounds for span\((P^{2M-1} \times P^{2N-1})\), where \(r = \nu(M) \geq 4\) and \(\nu(N) \leq r\) using the BP-cohomology, which are better than previously known bounds. Also, numerical examples are given.