Jacobi--Tsankov manifolds which are not 2-step nilpotent

Abstract

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.