We study when Calabi-Yau supermanifolds M with one complex bosonic coordinate and two complex fermionic coordinates are super Ricci-flat, and find that if the bosonic manifold is compact, it must have constant scalar curvature. In [1], we found that super Ricci-flat Kähler manifolds with one fermionic dimension and an arbitrary number of bosonic dimensions exist above a bosonic manifold with a ...

We consider Bel–Robinson-like higher derivative conserved two-index tensors H µν in simple matter models, following a recently suggested Maxwell field version. In flat space, we show that they are essentially equivalent to the true stress-tensors. In curved Ricci-flat backgrounds it is possible to redefine H µν so as to overcome non-commutativity of covariant derivatives, and maintain conservat...

Abstract We give a complete classification of left invariant para-Kähler structures on four-dimensional simply connected Lie groups up to an automorphism. As application we discuss some curvatures properties the canonical connection associated these as flat, Ricci flat and existence solitons.