Abstract
We construct an example of a quasi-Kähler structure of cohomogeneity 1 on \( S^{2}\times S^{4} \).
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References
Borel A. and Serre J.-P., “Détermination des p-puissances réduites de Steenrod dans la cohomologie des groupes classiques,” C. R. Acad. Sci. Paris, vol. 233, 680–682 (1951).
Calabi E. and Eckmann B., “A class of compact, complex manifolds which are not algebraic,” Ann. Math., vol. 58, no. 3, 494–500 (1953).
Datta B. and Subramanian S., “Nonexistence of almost complex structures on products of even-dimensional spheres,” Topol. Appl., vol. 36, no. 1, 39–42 (1990).
Smolentsev N. K., “On almost complex structures on products of six-dimensional spheres,” Uch. Zap. Kazan. Gos. Univ. Ser. Fiz.-Mat. Nauki, vol. 151, no. 4, 116–135 (2004).
Böhm C., “Inhomogeneous Einstein metrics on low-dimensional spheres and other low-dimensional spaces,” Invent. Math., vol. 134, no. 1, 145–176 (1998).
Grove K. and Ziller W., “Curvature and symmetry of Milnor spheres,” Ann. Math., vol. 152, no. 1, 331–367 (2000).
Foscolo L. and Haskins M., “New \( G_{2} \)-holonomy cones and exotic nearly Kähler structures on \( S^{6} \) and \( S^{3}\times S^{3} \),” Ann. Math., vol. 185, no. 1, 59–130 (2017).
Bredon G. E.,Introduction to Compact Transformation Groups, Acad. Press, New York (1972).
Montgomery D. and Samelson H., “Transformation groups on spheres,” Ann. Math., vol. 44, 454–470 (1943).
Borel A., “Les bouts des espaces homogènes de groupes de Lie,” Ann. Math., vol. 58, 443–457 (1953).
Poncet J., “Groupes de Lie compacts de transformations de l’espace euclidien et les sphères comme espaces homogènes,” Comment. Math. Helv., vol. 33, 109–120 (1959).
Gray A. and Hervella L. M., “The sixteen classes of almost Hermitian manifolds and their linear invariants,” Ann. Mat. Pura Appl., vol. 123, 35–58 (1980).
Nagy N.-A., “Nearly Kähler geometry and Riemannian foliations,” Asian J. Math., vol. 3, 481–504 (2002).
Butruille J.-B., “Classification des variétés approximativement kähleriennes homogénes,” Ann. Global Anal. Geom., vol. 27, 201–225 (2005).
Podestà F. and Spiro A., “Six-dimensional nearly Kähler manifolds of cohomogeneity one,” J. Geom. Phys., vol. 60, no. 2, 156–164 (2010).
Podestà F. and Spiro A., “Six-dimensional nearly Kähler manifolds of cohomogeneity one. II,” Comm. Math. Phys., vol. 312, no. 2, 477–500 (2012).
Salamon S., “Almost parallel structures,” Contemp. Math., Global Diff. Geom.: The Math. Legacy of A. Gray, vol. 288, 162–181 (2001).
Hitchin N., “Stable forms and special metrics,” in: Global Differential Geometry: The Mathematical Legacy of Alfred Gray. Proceedings of the international congress on differential geometry held in memory of Professor Alfred Gray, Bilbao, Spain, September 18–23, 2000, Amer. Math. Soc., Providence (2001), 70–89.
Hoelscher C. A., “Classification of cohomogeneity one manifolds in low dimensions,” Pacific J. Math., vol. 246, no. 1, 129–186 (2010).
Hoelscher C. A., “Diffeomorphism type of six-dimensional cohomogeneity one manifolds,” Ann. Global Anal. Geom., vol. 38, no. 1, 1–9 (2010).
Conti D. and Salamon S., “Generalized Killing spinors in dimension 5,” Trans. Amer. Math. Soc., vol. 359, no. 11, 5319–5343 (2007).
Eschenburg J.-H. and Wang M. Y., “The initial value problem for cohomogeneity one Einstein metrics,” J. Geom. Anal., vol. 10, no. 1, 109–137 (2000).
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The author was supported by the State Maintenance Program for the Leading Scientific Schools (Grant NSh–5913.2018.1).
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Daurtseva, N.A. Quasi-Kähler Structures of Cohomogeneity 1 on \( S^{2}\times S^{4} \). Sib Math J 61, 600–609 (2020). https://doi.org/10.1134/S0037446620040047
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DOI: https://doi.org/10.1134/S0037446620040047