Vector bundles over Grassmannians and the spectral geometry of the Riemann tensor

Methods from algebraic topology are often used to relate the algebraic properties of the Riemann curvature tensor to the geometry and topology of the underlying manifold. This paper provides a study of vector bundles over Grassmannians suitable for analyzing the spectral geometry of the Riemann tensor. Primarily, we study bundles over Gr k ( m ) , k ⩾ 3 , which are sub-bundles of the trivial bundle of rank m.