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

2002/05/08 by Peter Gilkey, Gilkey, Peter, Tan Zhang +1
Mathematics · Medicine · #53B20 #Advanced Neuroimaging Techniques and Applications #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Tensor decomposition and applications #math.DG #msc:53B20

paper · pdf · doi:10.48550/arxiv.math/0205080

arxiv created 2002/05/08 · openalex publication_date 2002/05/08 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let R be an algebraic curvature tensor for a non-degenerate inner product of signature(p,q) where q>4. 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(-) 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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