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PARTIAL CLASSIFICATION RESULTS FOR POSITIVE QUATERNION KÄHLER MANIFOLDS

MANUEL AMANN

Fachbereich Mathematik und Informatik, Westfälische Wilhelms-Universität Münster, Einsteinstr. 62, 48149 Münster, Germany

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https://doi.org/10.1142/S0129167X12500383Cited by:1 (Source: Crossref)

Abstract

Positive Quaternion Kähler Manifolds are Riemannian manifolds with holonomy contained in Sp(n)Sp(1) and with positive scalar curvature. Conjecturally, they are symmetric spaces. We prove this conjecture in dimension 20 under additional assumptions and we provide recognition theorems for the real Grassmannian in almost all dimensions.

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AMSC: 53C26

References

M. Amann, Positive Quaternion Kähler Manifolds, Ph.D. thesis, WWU Münster (2009), http://miami.uni-muenster.de/servlets/DocumentServlet?id=4869 . Google Scholar
M. Amann and A. Dessai, A note on the Â-genus for π2-finite manifolds with 𝕊1-symmetry, to appear in C. R. Math., doi:10.1016/j.crma.2010.01.011 . Google Scholar
M. Atiyah and F. Hirzebruch, Essays on Topology and Related Topics (Mémoires dédiés à Georges de Rham (Springer, New York, 1970) pp. 18–28. Crossref, Google Scholar
A. L. Besse , Einstein Manifolds , Classics in Mathematics ( Springer-Verlag , Berlin , 2008 ) . Google Scholar
A. Borel and J. De Siebenthal, Comment. Math. Helv. 23, 200 (1949), DOI: 10.1007/BF02565599. Crossref, Google Scholar
A. Borel and F. Hirzebruch, Amer. J. Math. 81, 315 (1959), DOI: 10.2307/2372747. Crossref, Web of Science, Google Scholar
T. Bröcker and T. tom Dieck , Representations of Compact Lie Groups , Graduate Texts in Mathematics 98 ( Springer-Verlag , New York , 1995 ) . Google Scholar
A. Dancer and A. Swann, Internat. J. Math. 8(5), 595 (1997), DOI: 10.1142/S0129167X97000317. Link, Web of Science, Google Scholar
A. Dancer and A. Swann, Internat. J. Math. 10(5), 541 (1999). Link, Web of Science, Google Scholar
F. Fang, Math. Res. Lett. 15(4), 641 (2008). Crossref, Web of Science, Google Scholar
A. Fujiki, Algebraic Geometry, Advanced Studies in Pure Mathematics 10 (North-Holland, Amsterdam, 1987) pp. 105–165. Crossref, Google Scholar
A. Gray, Michigan Math. J. 16, 125 (1969). Web of Science, Google Scholar
R. Herrera, Q. J. Math. 57(2), 203 (2006), DOI: 10.1093/qmath/hai002. Crossref, Web of Science, Google Scholar
H. Herrera and R. Herrera, J. Differential Geom. 61(3), 341 (2002). Crossref, Web of Science, Google Scholar
H. Herrera and R. Herrera, Erratum to "Â-genus on non-spin manifolds with S1 actions and the classification of positive quaternion-Kähler 12-manifolds", to appear . Google Scholar
S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Wiley Classics Library, Vol. I (John Wiley & Sons, New York, 1996). Reprint of the 1963 original, A Wiley-Interscience Publication . Google Scholar
A. Kollross, A classification of hyperpolar and cohomogeneity one actions, Ph.D. thesis, Universität Augsburg (1998) . Google Scholar
C. LeBrun, Internat. J. Math. 6(3), 419 (1995). Link, Web of Science, Google Scholar
C. LeBrun and S. Salamon, Invent. Math. 118(1), 109 (1994), DOI: 10.1007/BF01231528. Crossref, Web of Science, Google Scholar
T. Nagano and M. Takeuchi, Proc. Japan Acad. Ser. A Math. Sci. 59(8), 384 (1983), DOI: 10.3792/pjaa.59.384. Crossref, Google Scholar
S. Salamon, Invent. Math. 67(1), 143 (1982), DOI: 10.1007/BF01393378. Crossref, Web of Science, Google Scholar
S. Salamon, Differential Geometry and its Applications, Mathematical Publications 1 (Silesian University Opava, Opava, 1993) pp. 387–404. Google Scholar
S. Salamon, Surveys in Differential Geometry: Essays on Einstein Manifolds, Surveys in Differential Geometry VI (International Press, Boston, MA, 1999) pp. 83–121. Google Scholar
U. Semmelmann and G. Weingart, J. Geom. Anal. 14(1), 151 (2004), DOI: 10.1007/BF02921870. Crossref, Web of Science, Google Scholar