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Super differential geometry. (English) Zbl 0587.58014
Rep., Akad. Wiss. DDR, Inst. Math. R-MATH-05/84, 187 p. (1984).
This is a lecture on supergeometry close to the approach of B. Kostant [Lect. Notes Math. 570, 177-306 (1977; Zbl 0358.53024)]. In particular all considered manifolds and geometrical objects are smooth (analytic supermanifolds are absent), supermanifolds are defined in terms of sheaves (there are no other supermanifolds in the sense of A. Rogers) and the theory of Lie superalgebras is not used. On the other hand the author gives us a review of results of E. German ”supermathematicians” concerning super fibre bundles. The main point of this lecture is a series of reduction theorems which state bijectivity of functors between categories of ”super” objects and categories of relevant traditional differential geometry objects. Such theorems are proved for functors between supermanifolds and Grassmann bundles (the Batherol theorem) super fibre bundles and certain 2nd degree (composed vector bundles, super G- bundles, G is a supergroup) and principal fibre bundles and then locally free O-modules (O is a supermanifold) and graded vector bundles. Moreover the author describes super Lie groups (the general definition is in terms of categories and an explicit example of the super linear group is given) and presents the tensor calculus on supermanifolds and the related de Rham complex. The author did not say exactly that in the case of nonparacompact base spaces all the reduction theorems fail (this is the reason for the separability assumption in 2.4, p. 38).
Reviewer: J.Czyz

MSC:
58A99 General theory of differentiable manifolds
55R15 Classification of fiber spaces or bundles in algebraic topology
18F15 Abstract manifolds and fiber bundles (category-theoretic aspects)
53C99 Global differential geometry