Flat pseudo-Euclidean Leibniz superalgebras

In this paper, we introduce pre-Lie and pre-Leibniz superalgebras, which generalize pre-Lie and pre-Leibniz algebras to the super setting. Additionally, we define a Levi-Civita product associated with a symmetric non-degenerate bilinear form on a non-associative superalgebra. This leads to the definition of flat pseudo-Euclidean left Leibniz superalgebras as those whose Levi-Civita product induces a pre-Leibniz structure. We study the structure of flat pseudo-Euclidean left Leibniz superalgebras and provide a characterization theorem. In the second part, we focus on quadratic Leibniz superalgebras and show that such a superalgebra is flat if and only if it is symmetric Leibniz and 2-step nilpotent. We further study the structure of quadratic 2-step nilpotent symmetric Leibniz superalgebras. Finally, we introduce the notion of double extension for flat pseudo-Euclidean (resp. Lie) left Leibniz superalgebras and prove that any flat pseudo-Euclidean non-Lie left Leibniz superalgebra can be obtained by a sequence of double extensions starting from a flat pseudo-Euclidean Lie superalgebra.