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Two supergeometries from four-fold gradings of superalgebras. (English) Zbl 0599.53065
The aim of the paper is to generalize the symmetric Riemannian pair to a symmetric super Riemannian quadruple in supergeometry. The super Riemannian quadruple can be endowed either with a $${\mathbb{Z}}_ 4$$ or $${\mathbb{Z}}_ 2\times {\mathbb{Z}}_ 2$$-graded structure. Using the supersymmetric Cartan-Maurer equations for supergroups, we introduce two extensions of the Cartan structure equations, describing two types of supergeometries corresponding to $${\mathbb{Z}}_ 4$$ and $${\mathbb{Z}}_ 2\times {\mathbb{Z}}_ 2$$ gradings of the Lie superalgebra of the superisometries. Such supergeometries can describe the supersymmetric unification of two Riemannian geometries, describing space-time and internal degrees of freedom.
##### MSC:
 53C80 Applications of global differential geometry to the sciences 83E50 Supergravity 17A70 Superalgebras