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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.

58A50 Supermanifolds and graded manifolds
53C29 Issues of holonomy in differential geometry
32C11 Complex supergeometry
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[1] Batchelor, M.: The structure of supermanifolds. Trans. Am. Math. Soc. 253, 329–338 (1979) · Zbl 0413.58002
[2] Besse, A.L.: Einstein Manifolds. Springer, Berlin-Heidelberg-New York (1987) · Zbl 0613.53001
[3] Bryant, R.: Recent advances in the theory of holonomy. Séminaire Bourbaki 51 éme année. 1998–99. no 861
[4] Cortés, V.: A new construction of homogeneous quaternionic manifolds and related geometric structures. Mem. Am. Math. Soc. 147(700), viii+63 (2000) · Zbl 1037.53030
[5] Cortés, V.: Odd Riemannian symmetric spaces associated to four-forms. Math. Scand. 98(2), 201–216 (2006) · Zbl 1184.53055
[6] Deligne, P., Morgan, J.W.: Notes on supersymmetry (following Joseph Bernstein). In: Quantum Fields and Strings: A Course for Mathematicians, Vols. 1, 2, Princeton, NJ, 1996/1997, pp. 41–97. Am. Math. Soc., Providence (1999) · Zbl 1170.58302
[7] Galaev, A., Leistner, T.: Recent developments in pseudo-Riemannian holonomy theory. In: Handbook of Pseudo-Riemannian Geometry. IRMA Lectures in Mathematics and Theoretical Physics (2009, to appear) · Zbl 1214.53004
[8] Goertsches, O.: Riemannian supergeometry. Math. Z. 260(3), 557–593 (2008) · Zbl 1154.58001
[9] Joyce, D.: Compact Manifolds with Special Holonomy. Oxford University Press, London (2000) · Zbl 1027.53052
[10] Joyce, D.: Riemannian Holonomy Groups and Calibrated Geometry. Oxford University Press, London (2007) · Zbl 1200.53003
[11] Kac, V.G.: Lie superalgebras. Adv. Math. 26, 8–96 (1977) · Zbl 0366.17012
[12] Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry, vol. 1. Interscience Wiley, New York (1963) · Zbl 0119.37502
[13] Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry, vol. 2. Interscience Wiley, New York (1967) · Zbl 0119.37502
[14] Manin, Y.I.: Gauge Field Theory and Complex Geometry. Grundlehren, vol. 289. Springer, Berlin (1988). First appeared as Kalibrovochnye polya i kompleksnaya geometriya, Nauka, Moscow (1984) · Zbl 0576.53002
[15] Leites, D.A.: Introduction to the theory of supermanifolds. Usp. Mat. Nauk 35(1), 3–57 (1980). Translated in Russ. Math. Surv. 35(1), 1–64 (1980) · Zbl 0439.58007
[16] Leites, D.A.: Theory of Supermanifolds. Petrozavodsk (1983) (in Russian) · Zbl 0599.58001
[17] Leites, D.A., Poletaeva, E., Serganova, V.: On Einstein equations on manifolds and supermanifolds. J. Nonlinear Math. Phys. 9(4), 394–425 (2002) · Zbl 1009.35085
[18] Leites, D.A. (ed.): Supersymmetries. Algebra and Calculus. Springer (2009, to appear)
[19] Poletaeva, E.: Analogues of Riemann tensors for the odd metric on supermanifolds. Acta Appl. Math. 31(2), 137–169 (1993) · Zbl 0795.53025
[20] Poletaeva, E.: The analogs of Riemann and Penrose tensors on supermanifolds. arXiv:math/0510165 · Zbl 0795.53025
[21] Schwachhöfer, L.J.: Connections with irreducible holonomy representations. Adv. Math. 160(1), 1–80 (2001) · Zbl 1037.53035
[22] Serganova, V.V.: Classification of simple real Lie superalgebras and symmetric superspaces. Funct. Anal. Appl. 17(3), 200–207 (1983) (in Russian) · Zbl 0545.17001
[23] Varadarajan, V.S.: Supersymmetry for Mathematicians: An Introduction. Courant Lecture Notes, vol. 11. Am. Math. Soc., Providence (2004) · Zbl 1142.58009
[24] Wu, H.: Holonomy groups of indefinite metrics. Pac. J. Math. 20, 351–382 (1967) · Zbl 0149.39603
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