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Lectures on n-Sasakian Manifolds

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Geometry of Manifolds with Non-negative Sectional Curvature

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2110))

Abstract

These notes grew from a series of four 90 min lectures given at CIMAT in Guanajuato, Mexico in December 2010. They aim to present contemporary and historical topics in the theory of isoparametric hypersurfaces and discuss related recently published examples of CR submanifolds of complex projective space. Though we assume little to no background these notes should also hold the interest of an expert. The notion of an n-Sasakian manifold is a generalisation of the notion of a 3-Sasakian manifold. The examples discussed are related to the theory of isoparametric hypersurfaces of spheres with four principal curvatures. These examples carry Einstein metrics and in some special cases carry metrics with positive sectional curvature.

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References

  1. K. Abe, Applications of a Riccati type differential equation to Riemannian manifolds with totally geodesic distributions. Tôhoku Math. J. 25, 425–444 (1973)

    Article  MATH  Google Scholar 

  2. U. Abresch, Isoparametric hypersurfaces with four or six distinct principal curvatures. Necessary conditions on the multiplicities. Math. Ann. 264, 283–302 (1983)

    MATH  MathSciNet  Google Scholar 

  3. S. Aloff, N. Wallach, An infinite family of distinct 7-manifolds admitting positively curved Riemannian structures. Bull. Am. Math. Soc. 81, 93–97 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  4. M. Berger, Les variétés riemanniennes homogènes normales simplement connexes à courbure strictement positive. Ann. Scuola Norm. Sup. Pisa (3) 15, 179–246 (1961)

    Google Scholar 

  5. A.L. Besse, Einstein Manifolds. Reprint of the 1987 edition. Classics in Mathematics (Springer, Berlin, 2008)

    Google Scholar 

  6. C. Boyer, K. Galicki, 3-Sasakian manifolds, in Surveys in Differential Geometry VI. Essays on Einstein Manifolds, ed. by C. LeBrun, M. Wang (International Press, Boston, 1999), pp. 123–184

    Google Scholar 

  7. C. Boyer, K. Galicki, Sasakian Geometry. Oxford Mathematical Monographs (Oxford University Press, Oxford, 2008)

    MATH  Google Scholar 

  8. E. Cartan, Sur des familles remarquables d’hypersurfaces isoparamétriques dans les espaces sphériques. Math. Z. 45, 335–367 (1939)

    Article  MathSciNet  Google Scholar 

  9. T. Cecil, Q.S. Chi, G. Jensen, Isoparametric hypersurfaces with four principal curvatures. Ann. Math. (2) 166, 1–76 (2007)

    Google Scholar 

  10. Q.S. Chi, A note on the paper, “Isoparametric hypersurfaces with four principal curvatures”. Hongyou Wu Memorial Volume. Pac. J. Appl. Math. 3, 127–134 (2011)

    MATH  MathSciNet  Google Scholar 

  11. Q.S. Chi, Isoparametric hypersurfaces with four principal curvatures, II. Nagoya Math. J. 204, 1–18 (2011)

    MATH  MathSciNet  Google Scholar 

  12. Q.S. Chi, Isoparametric hypersurfaces with four principal curvatures revisited. Nagoya Math. J. 193, 129–154 (2009)

    MATH  MathSciNet  Google Scholar 

  13. Q.S. Chi, Isoparametric hypersurfaces with four principal curvatures, III. J. Differential Geom. 94, 469–540 (2013)

    MATH  MathSciNet  Google Scholar 

  14. O. Dearricott, Positive sectional curvature on 3-Sasakian manifolds. Ann. Global Anal. Geom. 25, 59–72 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  15. O. Dearricott, n-Sasakian manifolds. Tohoku Math. J. (2) 60, 329–347 (2008)

    Google Scholar 

  16. O. Dearricott, A 7-manifold with positive curvature. Duke Math. J. 158, 307–346 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  17. J. Dorfmeister, E. Neher, An algebraic approach to isoparametric hypersurfaces in spheres. I, II. Tôhoku Math. J. (2) 35, 18–224 (1983)

    Google Scholar 

  18. J. Dorfmeister, E. Neher, Isoparametric hypersurfaces, case g = 6,  m = 1. Commun. Algebra 13, 2299–2368 (1985)

    Google Scholar 

  19. J. Dorfmeister, E. Neher, Isoparametric triple systems with special \(\mathbb{Z}\)-structure. Algebras Groups Geom. 7, 21–94 (1990)

    MATH  MathSciNet  Google Scholar 

  20. D. Ferus, H. Karcher, H.F. Münzner, Cliffordalgebren und neue isoparametrische Hyperflächen. Math. Z. 177, 479–502 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  21. W. Hsiang, H.B. Lawson, Minimal submanifolds of low cohomogeneity. J. Differ. Geom. 5, 1–38 (1971)

    MATH  MathSciNet  Google Scholar 

  22. S. Immervoll, On the classification of isoparametric hypersurfaces with four distinct principal curvatures in spheres. Ann. Math. (2) 168, 1011–1024 (2008)

    Google Scholar 

  23. R. Miyaoka, The Dorfmeister-Neher theorem on isoparametric hypersurfaces. Osaka J. Math. 46, 695–715 (2009)

    MATH  MathSciNet  Google Scholar 

  24. R. Miyaoka, G 2-orbits and isoparametric hypersurfaces. Nagoya Math. J. 203, 175–189 (2011)

    MATH  MathSciNet  Google Scholar 

  25. P. Molino, Riemannian foliations (Translated from the French by Grant Cairns). Progress in Mathematics, vol. 73 (Birkhäuser, Boston, 1988)

    Google Scholar 

  26. H.F. Münzner, Isoparametrische Hyperflächen in Sphären. Math. Ann. 251, 57–71 (1980)

    Article  MATH  Google Scholar 

  27. B. O’Neill, The fundamental equations of a submersion. Michigan Math. J. 13, 459–469 (1966)

    Article  MATH  MathSciNet  Google Scholar 

  28. H. Ozeki, M. Takeuchi, On some types of isoparametric hypersurfaces in spheres. I. Tôhoku Math. J. (2) 27, 515–559 (1975)

    Google Scholar 

  29. S. Stolz, Multiplicities of Dupin hypersurfaces. Invent. Math. 138, 253–279 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  30. K. Yano, M. Kon, CR Submanifolds of Kaehlerian and Sasakian Manifolds. Progress in Mathematics, vol. 30 (Birkhäuser, Boston, 1983)

    Google Scholar 

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Acknowledgements

Thanks go to Daniele Grandini for his reading over drafts, Quo-Shin Chi for his suggestions.

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  1. Centro de Investigación en Matemt́icas, Guanajuato, Mexico

    Owen Dearricott

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  1. Owen Dearricott
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Dearricott, O. (2014). Lectures on n-Sasakian Manifolds. In: Geometry of Manifolds with Non-negative Sectional Curvature. Lecture Notes in Mathematics, vol 2110. Springer, Cham. https://doi.org/10.1007/978-3-319-06373-7_4

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