2006/09/20 by Miguel Brozos‐Vázquez, M. Brozos-Vazquez, Peter Gilkey +3 · 1 citation
Mathematics · Physics and Astronomy · #53C20 #Advanced Differential Geometry Research #Black Holes and Theoretical Physics #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #math.DG #msc:53C20
paper · pdf · doi:10.48550/arxiv.math/0609565
arxiv created 2006/09/20 · openalex publication_date 2006/09/20 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
An algebraic curvature tensor A is said to be Jacobi-Tsankov if J(x)J(y)=J(y)J(x) for all x,y. This implies J(x)J(x)=0 for all x; necessarily A=0 in the Riemannian setting. Furthermore, this implies J(x)J(y)=0 for all x,y if the dimension is at most 13. We exhibit a 14-dimensional algebraic curvature tensor in signature (8,6) which is Jacobi--Tsankov but which has J(x)J(y) non 0 for some x,y. We determine the group of symmetries of this tensor and show that it is geometrically realizable by a wide variety of pseudo-Riemannian manifolds which are geodesically complete and have vanishing scalar Weyl invariants. Some of the manifolds in the family are symmetric spaces. Some are 0-curvature homogeneous but not locally homogeneous.