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Riemannian, Symplectic and Weak Holonomy

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Abstract

Much of the early work of Alfred Gray was concerned with the investigation of Riemannian manifolds with special holonomy, one of the most vivid fields of Riemannian geometry in the 1960s and the following decades. It is the purpose of the present article to give a brief summary and an appreciation of Gray's contributions in this area on the one hand, and on the other hand to describe some of the more recent developments in the theory of non-Riemannian or,more specifically, symplectic holonomy groups. Namely, we show that the Merkulov twistor space of a connection on a symplectic manifold M whose holonomy group is irreducible and properly contained in Sp(V) consists of maximal totally geodesic Lagrangian submanifolds of M.

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References

  1. Alekseevskii, D. V.: Compact quaternion spaces, Functional Anal. Appl. 2(1968), 106–114.

    Google Scholar 

  2. Alekseevskii, D. V. and Cortés, V.: Classification of stationary compact homogeneous special pseudo Kähler manifolds of semisimple group, Preprint, 1998.

  3. Ambrose, W. and Singer, I. M.: A Theorem on holonomy, Trans. Amer. Math. Soc. 75(1953), 428–443.

    Google Scholar 

  4. Berger, M.: Sur les groupes d'holonomie des variétés à connexion affine et des variétés Riemanniennes, Bull. Soc. Math. France 83(1955), 279–330.

    Google Scholar 

  5. Berger, M.: Les espaces symétriques noncompacts, Ann. Sci. École Norm. Sup. 74(1957), 85–177.

    Google Scholar 

  6. Besse, A. L.: Einstein Manifolds, Ergeb. Math. Grenzgeb. (3) 10, Springer-Verlag, Berlin, 1987.

  7. Borel, A.: Some remarks about Lie groups transitive on spheres and tori, Bull. Amer.Math. Soc. 55(1949), 580–587.

    Google Scholar 

  8. Borel, A.: Le plan projectif des octaves et les sphères comme espaces homogènes, C.R. Acad. Sci. Paris 230(1950), 1378–1380.

    Google Scholar 

  9. Borel, A. and Lichnerowicz, A.: Groupes d'holonomie des variétés riemanniennes, C.R. Acad. Sci. Paris 234(1952), 1835–1837.

    Google Scholar 

  10. Brown, R. and Gray, A.: Riemannian manifolds with holonomy group Spin (9), in Differential Geometry in Honor of K. Yano, Kinokuniya, Tokyo, 1972, pp. 41–59.

  11. Bryant, R.: Metrics with exceptional holonomy, Ann. Math. 126(1987), 525–576.

    Google Scholar 

  12. Bryant, R.: Two exotic holonomies in dimension four, path geometries, and twistor theory, Proc. Sympos. Pure Math. 53(1991), 33–88.

    Google Scholar 

  13. Bryant,.: Classical, exceptional, and exotic holonomies: A status report, in Actes de la Table Ronde de Géométrie Différentielle en l'Honneur de Marcel Berger, Collection SMF Séminaires and Congrès 1 (Soc. Math. de France), 1996, pp. 93–166.

  14. Calabi, E.: On Kähler manifolds with vanishing canonical class, in Algebraic Geometry and Topology, A Symposium in Honor of S. Lefschetz, Princeton Univ. Press, London, 1955.

    Google Scholar 

  15. Calabi, E.: Métriques kähleriennes et fibrés holomorphes, Ann. Sci. École Norm. Sup. 12(1979), 269–294.

    Google Scholar 

  16. Cartan, É.: Les groupes de transformations continus, infinis, simples, Ann. Sci. École Norm. Sup. 26(1909), 93–161.

    Google Scholar 

  17. Cartan, É.: Sur les variétés à connexion affine et la théorie de la relativité généralisée I & II, Ann. Sci. École Norm. Sup. 40(1923), 325–412 and 41 (1924), 1–25 or Oeuvres complètes, Vol. III, 659–746 and 799–824.

    Google Scholar 

  18. Cartan, É.: Sur une classe remarquable d'espaces de Riemann, Bull. Soc. Math. France 54(1926), 214–264 and 55 (1927), 114–134 or Oeuvres complètes, Vol. I(2), 587–659.

    Google Scholar 

  19. Cartan, É.: Les groupes d'holonomie des espaces généralisés, Acta Math. 48(1926), 1–42 or Oeuvres complètes Vol. III(2), 997–1038.

    Google Scholar 

  20. Chi, Q.-S.: Symplectic connections and exotic holonomy, Internat. J. Math. 9(1998), 47–61.

    Google Scholar 

  21. Chi, Q.-S., Merkulov, S. A. and Schwachhöfer, L. J.: On the existence of infinite series of exotic holonomies, Invent. Math. 126(1996), 391–411.

    Google Scholar 

  22. Chi, Q.-S., Merkulov, S. A. and Schwachhöfer, L. J.: Exotic holonomies E (a)7 , Internat. J.Math. 8(1997), 583–594.

    Google Scholar 

  23. Chi, Q.-S. and Schwachhöfer, L. J.: Exotic holonomy on moduli spaces of rational curves, Differential Geom. Appl. 8(1998), 105–134.

    Google Scholar 

  24. DeRham, G.: Sur la réductibilité d'un espace de Riemann, Comment. Math. Helv. 26, 328–344 (1952)

    Google Scholar 

  25. Gray, A.: Nearly Kähler manifolds, J. Differential Geom. 4(1970), 283–310.

    Google Scholar 

  26. Gray, A.: Weak holonomy groups, Math. Z. 123(1971), 290–300.

    Google Scholar 

  27. Gray, A.: A note on manifolds whose holonomy is a subgroup of Sp.(n) Sp (1), Michigan Math. J. 16(1965), 125–128.

    Google Scholar 

  28. Hano, J. and Ozeki, H.: On the holonomy groups of linear connections, Nagoya Math. J. 10(1956), 97–100.

    Google Scholar 

  29. Joyce, D.: Compact Riemannian 7–manifolds with holonomy G2: I & II, J. Differential Geom. 43(1996), 291–375.

    Google Scholar 

  30. Kobayashi, S. and Nagano, K.: On filtered Lie algebras and geometric structures II, J. Math. Mech. 14, 513–521 (1965)

    Google Scholar 

  31. Kobayashi, S. and Nomizu, K.: Foundations of Differential Geometry, Vols. 1 & 2, Wiley-Interscience, New York, 1963.

    Google Scholar 

  32. Merkulov, S. A.: Moduli of compact complex Legendre submanifolds of complex contact manifolds, Math. Res. Lett. 1(1994), 717–727.

    Google Scholar 

  33. Merkulov, S. A.: Existence and geometry of Legendre moduli spaces, Math. Z. 226(1997), 211–265.

    Google Scholar 

  34. Merkulov, S. A.: Moduli spaces of compact complex submanifolds of complex fibered manifolds, Math. Proc. Cambridge Philos. Soc. 118(1995), 71–91.

    Google Scholar 

  35. Merkulov, S. A.: Geometry of Kodaira moduli spaces, Proc. Amer. Math. Soc. 124(1996), 1499–1506.

    Google Scholar 

  36. Merkulov, S. A. and Schwachhöfer, L. J.: Classification of irreducible holonomies of torsion free affine connections, Ann. Math. 150(1999), 1177–1179.

    Google Scholar 

  37. Merkulov, S. A. and Schwachhöfer, L. J.: Twistor solution of the holonomy problem, in S. A. Hugget (ed.), The Geometric Universe, Science, Geometry and the Work of Roger Penrose, Oxford Univ. Press, Oxford, 1998, pp. 395–402.

    Google Scholar 

  38. Montgomery, D. and Samelson, H.: Transformation groups of spheres, Ann. Math. 44(1943), 454–470.

    Google Scholar 

  39. Montgomery, D. and Samelson, H.: Groups transitive on the n-dimensional torus, Bull. Amer. Math. Soc. 49(1943), 455–456.

    Google Scholar 

  40. Nijenhuis, A.: On the holonomy group of linear connections, Indag. Math. 15(1953), 233–249, 16 (1954), 17–25.

    Google Scholar 

  41. Nijenhuis, A.: A note on infinitesimal holonomy groups, Nagoya Math. J. 12(1957), 145–147.

    Google Scholar 

  42. Salamon, S.: Riemannian Geometry and Holonomy Groups, Pitman Res. Notes Math. Ser. 201, Longman Scientific & Technical, Essex, 1989.

    Google Scholar 

  43. Schwachhöfer, L. J.: On the classification of holonomy representations, Habilitationsschrift, Universität Leipzig, 1998.

  44. Simons, J.: On transitivity of holonomy systems, Ann. Math. 76(1962), 213–234.

    Google Scholar 

  45. Weinstein, A.: The local structure of Poisson manifolds, J. Differential Geom. 18(1983), 523–557.

    Google Scholar 

  46. Yau, S. T.: On the Ricci curvature of a compact Kähler mainfold and the complex Monge-Ampère equation I, Comm. Pure Appl. Math. 31(1978), 339–411.

    Google Scholar 

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  1. Mathematisches Institut, Universität Leipzig, Augustusplatz 10–11, 04109, Leipzig, Germany

    Lorenz J. Schwachhöfer

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Schwachhöfer, L.J. Riemannian, Symplectic and Weak Holonomy. Annals of Global Analysis and Geometry 18, 291–308 (2000). https://doi.org/10.1023/A:1006769532110

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