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

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

2002-05-08

arxiv.org/abs/math/0205080v1

Mathematics

Differential Geometry

Scientific paper

Let R be an algebraic curvature tensor for a non-degenerate inner product of signature(p,q) where q>4. If $\pi$ is a spacelike 2 plane, let $R(\pi)$ be the associated skew-symmetric curvature operator. We classify the algebraic curvature tensors so R(-) 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(-) is constant.

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