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Holonomy of supermanifolds. (English) Zbl 1182.58004
The author introduces and studies holonomy groups for connections on locally free sheaves on smooth supermanifolds. To define the holonomy algebra, he uses covariant derivatives of the curvature tensor and parallel displacements. The holonomy group is defined as a Lie supergroup. The infinitesimal holonomy algebra (which coincides with the usual holonomy algebra in the analytic case) is also defined. Then he shows that any parallel section of a sheaf is uniquely defined by its value at any point, despite the fact that this is not usually the case for a section of a sheaf over a supermanifold. Next, he establishes several important one-to-one correspondences: one between parallel sections and holonomy-invariant vectors (as in the usual case of vector bundles over smooth manifolds), another one between parallel locally direct subsheaves and holonomy-invariant vector supersubspaces and, finally, another one between parallel tensors on a supermanifold and holonomy-invariant tensors at one point. Several generalizations of notions from ordinary differential geometry to the case of supermanifolds are also presented: Levi-Civita connections, Kählerian, special Kählerian, hyper-Kählerian and quaternionic-Kählerian supermanifolds (which are characterized by their holonomy). Special Kählerian supermanifolds are Ricci-flat and, conversely, Ricci-flat simply connected Kählerian supermanifolds are special Kählerian. A generalization of the theorem of Wu to the case of Riemannian supermanifolds is obtained. Finally, the author introduces Berger superalgebras and furnishes many interesting examples in the complex case.

MSC:
58A50 Supermanifolds and graded manifolds
53C29 Issues of holonomy in differential geometry
32C11 Complex supergeometry
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