Basic geometry of G-supermanifolds (2024)

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  • Claudio Bartocci4,
  • Ugo Bruzzo4 &
  • Daniel Hernández-Ruipérez5

Part of the book series: Mathematics and Its Applications ((MAIA,volume 71))

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Abstract

The first five Sections of this Chapter, the technical core of this book, are dedicated to set down the basic differential geometry of G-supermanifolds, by introducing the fundamental objects one needs: morphisms, products, supervector bundles, and differential forms. It should be pointed out that the relevant definitions are quite different from the usual ones, and rather in the spirit of the algebraic geometry. This is a consequence of the fact that part of the information conveyed by the structure sheaf of a G-supermanifold is not otherwise embodied in the associated topological space.

They explore the new field and bring back their spoils — a few simple generalizations — to apply them to the practical world of three dimensions. Some guiding light will be given to the attempts to build a scheme of things entire A.S. Eddington

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References

  1. The fact that this morphism exists and is uniquely defined, albeit seemingly, is not entirely trivial; for a proof, see [Gro1].

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  2. Cf. Definition II.2.3.

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  3. The notion of body of a supermanifold is more general, and applies to a wider category of supermanifolds than DeWitt ones [BoyG, CaRT].

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  4. Since Gl(r) is not abelian, H1(X, Gl(r)) is not a group, but only a pointed set; see [Hirz].

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  5. Even though we shall not need this fact in the sequel, let us notice that Batchelor’s theorem (Corollary III.1.9) implies an isomorphism H1(X, Aut Λ Rr) ≃ H1(X, Gl(r)); a direct proof of this fact was given in [Bch1].

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  6. This because any graded ideal is contained in a maximal graded ideal. Proof of this statement, which makes use of Zorn’s lemma, goes as in the non-graded case [AtM].

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Authors and Affiliations

  1. Department of Mathematics, University of Genoa, Genoa, Italy

    Claudio Bartocci&Ugo Bruzzo

  2. Department of Pure and Applied Mathematics, University of Salamanca, Salamanca, Spain

    Daniel Hernández-Ruipérez

Authors

  1. Claudio Bartocci

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  2. Ugo Bruzzo

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  3. Daniel Hernández-Ruipérez

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© 1991 Springer Science+Business Media Dordrecht

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Bartocci, C., Bruzzo, U., Hernández-Ruipérez, D. (1991). Basic geometry of G-supermanifolds. In: The Geometry of Supermanifolds. Mathematics and Its Applications, vol 71. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-3504-7_4

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  • DOI: https://doi.org/10.1007/978-94-011-3504-7_4

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