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Bazaikin, Ya. V.

Noncompact Riemannian spaces with the holonomy group spin\((7)\) and 3-Sasakian manifolds. (English. Russian original) Zbl 1179.53057

Proc. Steklov Inst. Math. 263, 2-12 (2008); translation from Tr. Mat. Inst. Steklova 263, 6-17 (2008).
The author considers the existence of Riemannian metrics having the special holonomy group \(\text{Spin}(7)\) which smoothly resolve the standard cone metrics on non-compact manifolds and orbifolds related to 7-dimensional 3-Sasakian spaces. This completes his study began in [Sib. Mat. Zh. 48, No. 1, 11–32 (2007); translation in Sib. Math. J. 48, No. 1, 8–25 (2007; Zbl 1164.53360)].
Contents include: Introduction and main results (which contains an overview of the topic); Construction of \(\text{Spin}(7)\)-holonomy metrics; \(\text{Spin}(7)\)-holonomy metrics on \({\mathcal M}_1\); and References.
Reviewer: Joseph D. Zund (Las Cruces)

Cited in 6 Documents

MSC:

53C29 Issues of holonomy in differential geometry
53C27 Spin and Spin\({}^c\) geometry
53C10 \(G\)-structures

Keywords:

non-compact Riemannian spaces; holonomy group \(\text{Spin}(7)\); 3-Sasakian spaces

Citations:

Zbl 1164.53360
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Full Text: DOI

References:

[1] Ya. V. Bazaikin, ”On the New Examples of Complete Noncompact Spin(7)-Holonomy Metrics,” Sib. Mat. Zh. 48(1), 11–32 (2007) [Sib. Math. J. 48, 8–25 (2007)]. · Zbl 1164.53360
[2] R. L. Bryant, ”Metrics with Exceptional Holonomy,” Ann. Math. 126, 525–576 (1987). · Zbl 0637.53042 · doi:10.2307/1971360
[3] R. L. Bryant and S. L. Salamon, ”On the Construction of Some Complete Metrics with Exceptional Holonomy,” Duke Math. J. 58, 829–850 (1989). · Zbl 0681.53021 · doi:10.1215/S0012-7094-89-05839-0
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[5] M. Cvetič, G. W. Gibbons, H. Lü, and C. N. Pope, ”New Complete Non-compact Spin(7) Manifolds,” Nucl. Phys. B 620, 29–54 (2002). · Zbl 1037.53032 · doi:10.1016/S0550-3213(01)00559-4
[6] M. Cvetič, G. W. Gibbons, H. Lü, and C. N. Pope, ”New Cohomogeneity One Metrics with Spin(7) Holonomy,” J. Geom. Phys. 49, 350–365 (2004). · Zbl 1092.53024 · doi:10.1016/S0393-0440(03)00108-6
[7] M. Cvetič, G. W. Gibbons, H. Lü, and C. N. Pope, ”Cohomogeneity One Manifolds of Spin(7) and G 2 Holonomy,” Phys. Rev. D 65(10), 106004 (2002). · Zbl 1031.53076
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[9] H. Kanno and Y. Yasui, ”On Spin(7) Holonomy Metric Based on SU(3)/U(1),” J. Geom. Phys. 43, 293–309 (2002). · Zbl 1024.53035 · doi:10.1016/S0393-0440(02)00018-9
[10] H. Kanno and Y. Yasui, ”On Spin(7) Holonomy Metric Based on SU(3)/U(1). II,” J. Geom. Phys. 43, 310–326 (2002). · Zbl 1025.53025 · doi:10.1016/S0393-0440(02)00021-9
[11] A. Gray, ”Weak Holonomy Groups,” Math. Z. 123, 290–300 (1971). · Zbl 0222.53043 · doi:10.1007/BF01109983
[12] C. Boyer and K. Galicki, ”3-Sasakian Manifolds,” in Surveys in Differential Geometry: Essays on Einstein Manifolds (International Press, Boston, MA, 1999), Surv. Diff. Geom. 6, pp. 123–184. · Zbl 1008.53047
[13] F. Reidegeld, ”Spin(7)-Manifolds of Cohomogeneity One,” in Special Geometries in Mathematical Physics: Workshop, Kuehlungsborn, 2008 (in press). · Zbl 1335.53002
[14] L. Bérard-Bergery, ”Sur de nouvelles variétés riemanniennes d’Einstein,” Inst. Élie Cartan, Univ. Nancy 6, 1–60 (1982).
[15] D. N. Page and C. N. Pope, ”Inhomogeneous Einstein Metrics on Complex Line Bundles,” Class. Quantum Grav. 4(2), 213–225 (1987). · Zbl 0613.53020 · doi:10.1088/0264-9381/4/2/005
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