Algebraic curvature tensors whose skew-symmetric curvature operator has constant rank 2

Abstract

Let R be an algebraic curvature tensor for a non-degenerate inner product of signature (p,q) where q≥5. If π is a spacelike 2 plane, let R(π) be the associated skew-symmetric curvature operator. We classify the algebraic curvature tensors so R(ċ) has constant rank 2 and show these are geometrically realizable by hypersurfaces in flat spaces. We also classify the Ivanov–Petrova algebraic curvature tensors of rank 2; these are the algebraic curvature tensors of constant rank 2 such that the complex Jordan normal form of R(ċ) is constant.