Splitting 3-plane sub-bundles over the product of two real projective spaces

Abstract

Let α be a real vector bundle of fiber dimension three over the product IRP (m) × IRP (n) which splits as a Whitney sum of line bundles. We show that the necessary and sufficient conditions for α to embed as a sub-bundle of a certain family of vector bundles β of fiber dimension m + n is the vanishing of the last three Stiefel-Whitney classes of the virtual bundle0 β − α. Among the target bundles β we consider the tangent bundle. 1