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Riemannian supergeometry. (English) Zbl 1154.58001
First, the author presents the foundations of supergeometry, considering supermanifolds and their morphisms, the tangent sheaf and vector fields, Lie supergroups and their Lie superalgebras, actions and representations. Then, he develops a theory of Riemannian supermanifolds, studying connections and geodesics showing that parallel displacement is an isometry and that an isometry of a connected Riemannian supermanifold is determined by its value and its derivative at one point. Graded Killing fields, the isometry group, homogeneous superspaces and Riemannian symmetric superspaces are also studied. Among the results obtained there is the one which states that a Lie supergroup with a bi-invariant graded Riemannian metric is a Riemannian symmetric superspace. Finally, several interesting examples of series of Riemannian symmetric superspaces are presented.

MSC:
58A50 Supermanifolds and graded manifolds
53C22 Geodesics in global differential geometry
53C05 Connections (general theory)
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53B21 Methods of local Riemannian geometry
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