zbMATH — the first resource for mathematics

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.

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
Full Text: DOI arXiv
[1] Baguis P. and Stavracou T. (2002). Normal Lie subsupergroups and non-Abelian supercircles. Int. J. Math. Math. Sci. 30(10): 581–591 · Zbl 1072.17013
[2] Ballmann, W.: Automorphism groups. Lecture notes. http://www.math.uni-bonn.de/people/hwbllmnn/archiv/autmor00.ps , accessed on April 7, (2000)
[3] Batchelor M. (1979). The structure of supermanifolds. Trans. Am. Math. Soc. 253: 329–338 · Zbl 0413.58002
[4] Berger M. (1957). Les espaces symétriques non compacts. Ann. Sci. Ecole Norm. Sup. 74: 85–177 · Zbl 0093.35602
[5] Berezin F.A. (1987). Introduction to Superanalysis. Reidel, Dordrecht · Zbl 0659.58001
[6] Boyer C.P. and Sánchez O.A. (1980). Valenzuela, Lie supergroup actions on supermanifolds. Trans. Am. Math. Soc. 323: 151–175 · Zbl 0724.58005
[7] Cahen, M., Parker, M.: Pseudo-Riemannian symmetric spaces. Mem. Am. Math. Soc. 24(229), iv+108 pp (1980) · Zbl 0438.53057
[8] Cariñena, J.F., Figueroa H.: Hamiltonian versus Lagrangian formulation of supermechanics. J. Phys. A Math. Gen. 30, 2705–2724 · Zbl 0949.70016
[9] Cortés, V.: Odd Riemannian symmetric spaces associated to four-forms, to appear in Math. Scand
[10] DeWitt B. (1992). Supermanifolds, 2nd edn. Cambridge University Press, Cambridge · Zbl 0874.53055
[11] Deligne, P., Morgan, J.W.: Notes on supersymmetry (following Joseph Bernstein). Quantum Fields and Strings: A Course for Mathematicians, Vols. 1, 2 (Princeton, NJ, 1996/1997), pp. 41–97. American Mathematical Society, Providence (1999) · Zbl 1170.58302
[12] Kath, I., Olbrich, M.: On the structure of pseudo-Riemannian symmetric spaces. arXiv:math.DG/0408249 · Zbl 1190.53051
[13] Kostant, B.: Graded manifolds, graded Lie theory, and prequantization, Differential Geometrical Methods in Mathematical Physics (Proc. Sympos., Univ. Bonn, Bonn, 1975), Lecture Notes in Mathematics, vol. 570, pp. 177–306 Springer, Berlin (1977) · Zbl 0358.53024
[14] Koszul, J.L.: Graded manifolds and graded Lie algebras. In: Proceedings of the International Meeting on Geometry and Physics (Bologna), Pitagora, pp. 71–84 (1982)
[15] Leites, D.A.: Introduction to the theory of supermanifolds. Uspekhi Mat. Nauk 35(1), 3–57 (1980), translated in Russian Math. Surveys, 35(1), 1–64 (1980) · Zbl 0439.58007
[16] Manin, Yu.I.: Gauge Field Theory and Complex Geometry, Grundlehren. First appeared as Kalibrovochnye polya i kompleksnaya geometriya, Nauka, Moscow 1984, vol. 289. Springer, Heidelberg (1988) · Zbl 0576.53002
[17] Monterde, J., Muñoz Masqué, J.: Variational problems on graded manifolds. In: Gotay, M.J., Marsden, J.E., Moncrief, V. (eds) Proceedings of the 1991 Joint Summer Research Conference on Mathematical Aspects of Classical Field Theory, Seattle, Contemp. Math. vol. 132, pp. 551–571. American Mathematical Society, Providence (1991)
[18] Monterde J. and Sánchez-Valenzuela O.A. (1996). The exterior derivative as a Killing vector field. Israel J. Math. 93: 157–170 · Zbl 0853.58010
[19] Monterde J. and Sánchez-Valenzuela O.A. (1997). Graded metrics adapted to splittings. Israel J. Math. 99: 231–270 · Zbl 0887.58004
[20] Montgomery, D.L. Zippin: Topological Transformation Groups, 3rd Printing. Interscience, New York (1965)
[21] Onishchik A.L. (1994). Flag supermanifolds, their automorphisms and deformations. In: Sophus Lie Memorial Conference (Oslo 1992), pp 289–302. Scand. University Press, Oslo
[22] Petersen P. (1998). Riemannian Geometry. Springer, New York · Zbl 0914.53001
[23] Scheunert M. (1979). The Theory of Lie Superalgebras, Lecture Notes in Mathematics, vol. 716. Springer, Berlin · Zbl 0407.17001
[24] Schmitt, T.: Super differential geometry, Akademie der Wissenschaften der DDR, Institut für Mathematik, report R-MATH-05/84, Berlin (1984)
[25] Serganova V. (1983). Classification of real simple Lie superalgebras and symmetric superspaces. Funct. Anal. Appl. 17(3): 200–207 · Zbl 0545.17001
[26] Stavracou T. (1998). Theory of connections of graded principal bundles. Rev. Math. Phys. 10(1): 47–79 · Zbl 0920.58004
[27] Varadarajan, V.S.: Supersymmetry for Mathematicians: An Introduction. Courant Lecture Notes, vol. 11. American Mathematical Society, Providence (2004) · Zbl 1142.58009
[28] Zirnbauer M.R. (1996). Riemannian symmetric superspaces and their origin in random-matrix theory. J. Math. Phys. 37(10): 4986–5018 · Zbl 0871.58005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.