The nonexistence of rank 4 IP tensors in signature (1,3)

Abstract

Let V be a real vector space of dimension 4 with a nondegenerate symmetric bilinear form of signature (1,3). We show that there exists no algebraic curvature tensor R on V so that its associated skew-symmetric operator R(⋅) has rank 4 and constant eigenvalues on the Grassmannian of nondegenerate 2-planes in V.