Quaternionic Kaehler manifolds

Whitt, Lee

doi:10.1090/S0002-9947-1982-0662060-6

Quaternionic Kaehler manifolds
HTML articles powered by AMS MathViewer

by Lee Whitt

Trans. Amer. Math. Soc. 272 (1982), 677-692

DOI: https://doi.org/10.1090/S0002-9947-1982-0662060-6

PDF | Request permission

Abstract:

The topological classification of $4$- and $8$- (real) dimensional compact quaternionic Kaehler manifolds is given. There is only the torus in dimension 4. In dimension 8, there are 12 homeomorphism classes; representatives are given explicitly.

References

Louis Auslander, Four dimensional compact locally hermitian manifolds, Trans. Amer. Math. Soc. 84 (1957), 379–391. MR 83798, DOI 10.1090/S0002-9947-1957-0083798-3
L. Auslander and M. Kuranishi, On the holonomy group of locally Euclidean spaces, Ann. of Math. (2) 65 (1957), 411–415. MR 86341, DOI 10.2307/1970053
Ludwig Bieberbach, Über die Bewegungsgruppen der Euklidischen Räume, Math. Ann. 70 (1911), no. 3, 297–336 (German). MR 1511623, DOI 10.1007/BF01564500
S. S. Chern, Complex manifolds without potential theory, Van Nostrand Mathematical Studies, No. 15, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR 0225346
Shoshichi Kobayashi and Katsumi Nomizu, Foundations of differential geometry. Vol. I, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1996. Reprint of the 1963 original; A Wiley-Interscience Publication. MR 1393940
K. Kodaira, On Kähler varieties of restricted type (an intrinsic characterization of algebraic varieties), Ann. of Math. (2) 60 (1954), 28–48. MR 68871, DOI 10.2307/1969701

Lectures on general algebra

Serge Lang, Algebra, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965. MR 0197234
Morio Obata, Affine connections on manifolds with almost complex, quaternion or Hermitian structure, Jpn. J. Math. 26 (1956), 43–77. MR 95290, DOI 10.4099/jjm1924.26.0_{4}3
Andrew John Sommese, Quaternionic manifolds, Math. Ann. 212 (1974/75), 191–214. MR 425827, DOI 10.1007/BF01357140
Norman Steenrod, The topology of fibre bundles, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1999. Reprint of the 1957 edition; Princeton Paperbacks. MR 1688579
Joseph A. Wolf, Spaces of constant curvature, 2nd ed., University of California, Department of Mathematics, Berkeley, Calif., 1972. MR 0343213
Shing Tung Yau, Calabi’s conjecture and some new results in algebraic geometry, Proc. Nat. Acad. Sci. U.S.A. 74 (1977), no. 5, 1798–1799. MR 451180, DOI 10.1073/pnas.74.5.1798