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J. Geom. Phys. 91 (2015), 146–162. a Quaternion-Kähler Manifold Is a 4N Citations From References: 0 From Reviews: 0 MR3327056 (Review) 53C26 Gambioli, A. [Gambioli, Andrea] (3-DAWS-M); Nagatomo, Y. [Nagatomo, Yasuyuki] (J-MEIJ2); Salamon, S. [Salamon, Simon M.] (4-LNDKC) Special geometries associated to quaternion-K¨ahler8-manifolds. (English summary) J. Geom. Phys. 91 (2015), 146{162. A quaternion-K¨ahlermanifold is a 4n-dimensional smooth manifold (n ≥ 2) endowed with a Riemannian metric g with holonomy contained in the subgroup Sp(n)Sp(1) of SO(4n). In this interesting paper under review, the authors develop a calculus of differential forms on a quaternion-K¨ahlermanifold admitting an isometric circle action and establish the existence of direct links between quaternion-K¨ahlergeometry in dimension eight, half-flat geometry in dimension six, and G2 geometry in dimension seven. Gabriel Eduard V^ılcu References 1. B. Acharya, E. Witten, Chiral fermions from manifolds of G2 holonomy, hep- th/0106034. 2. A. Swann, Hyper-K¨ahlerand quaternionic K¨ahlergeometry, Math. Ann. 289 (1991) 421{450. MR1096180 3. R. Bryant, S. Salamon, On the construction of some complete metrics with excep- tional holonomy, Duke Math. J. 58 (1989) 829{850. MR1016448 4. K. Galicki, B. Lawson, Quaternionic reduction and quaternionic orbifolds, Math. Ann. 282 (1988) 1{21. MR0960830 5. N. Hitchin, A. Karlhede, U. Lindstr¨om,M. Roˇcek, Hyperk¨ahlermetrics and super- symmetry, Comm. Math. Phys. 108 (1987) 535{589. MR0877637 6. S. Salamon, Quaternionic K¨ahler manifolds, Invent. Math. 67 (1982) 143{171. MR0664330 7. F. Battaglia, Circle actions and Morse theory on quaternion-K¨ahler manifolds, J. Lond. Math. Soc. 59 (1999) 345{358. MR1688506 8. M. Atiyah, E. Witten, M-theory dynamics on a manifold of G2 holonomy, Adv. Theor. Math. Phys. 6 (2002) 1{106. MR1992874 9. M. Atiyah, J. Berndt, Projective planes, Severi varieties and spheres, in: Surveys in Differential Geometry, Int. Press, Somerville, 2003, pp. 1{27. MR2039984 10. V. Arnold, Relatives of the quotient of the complex projective plane by complex conjugation, Proc. Steklov Inst. Math. 224 (1999) 46{56. MR1721354 11. M. Pontecorvo, On twistor spaces of anti-self-dual Hermitian surfaces, Trans. Amer. Math. Soc. 331 (1992) 653{661. MR1050087 12. A. Haydys, HyperK¨ahler and quaternionic K¨ahlermanifolds with S1-symmetries, J. Geom. Phys. 58 (2008) 293{306. MR2394039 13. N.J. Hitchin, On the hyperk¨ahler/quaternionk¨ahlercorrespondence, Comm. Math. Phys. 324 (2013) 77{106. MR3116317 14. O. Macia, A. Swann, Twist geometry of the c-map, arXiv:1404.0785. cf. MR3324146 15. D. Conti, T.B. Madsen, Harmonic structures and intrinsic torsion, arXiv:1308.4083. 16. M.F. Atiyah, N.J. Hitchin, I.M. Singer, Self-duality in four-dimensional Riemannian geometry, Proc. R. Soc. Lond. Ser. A 362 (1978) 425{461. MR0506229 17. J. Eells, S. Salamon, Twistorial construction of harmonic maps of surfaces into four-manifolds, Ann. Sc. Norm. Super. Pisa 12 (1985) 589{640. MR0848842 18. S. Salamon, Harmonic and holomorphic maps, in: Geometry Seminar Luigi Bianchi, in: Lect. Notes Math., vol. 1164, Springer, 1985, pp. 161{224. MR0829230 19. M. Mamone Capria, S.M. Salamon, Yang{Mills fields on quaternionic spaces, Non- linearity 1 (1988) 517{530. MR0967469 20. K. Galicki, Y.S. Poon, Duality and Yang{Mills fields on quaternionic K¨ahlermani- folds, J. Math. Phys. 32 (1991) 1263{1268. MR1103479 21. Y. Nagatomo, T. Nitta, Vanishing theorem for quaternionic complexes, Bull. Lond. Math. Soc. 29 (1997) 359{366. MR1435574 22. J.A. Wolf, Complex homogeneous contact manifolds and quaternionic symmetric spaces, J. Math. Mech. 14 (1965) 1033{1047. See also Acta Math. 152 (1984) 141{ 142. MR0185554 23. A. Moroianu, U. Semmelmann, Clifford structures on Riemannian manifolds, Adv. Math. 228 (2011) 940{967. MR2822214 24. V.Y. Kraines, Topology of quaternionic manifolds, Trans. Amer. Math. Soc. 122 (1966) 357{367. MR0192513 25. R. Bryant, R. Harvey, Submanifolds in hyper-K¨ahlergeometry, J. Amer. Math. Soc. 2 (1989) 1{31. MR0953169 26. S. Salamon, Riemannian Geometry and Holonomy Groups, in: Pitman Research Notes in Mathematics, vol. 201, Longman, 1989. MR1004008 27. A. Swann, Some remarks on quaternion-Hermitian manifolds, Arch. Math. (Brno) 33 (1997) 349{354. MR1601349 28. D.V. Alekseevsky, Riemannian spaces with unusual holonomy groups, Funktsional. Anal. i Prilozhen. 2 (1968) 1{10. MR0231313 29. D.V. Alekseevsky, S. Marchiafava, M. Pontecorvo, Compatible almost complex structures on quaternion-K¨ahlermanifolds, Ann. Global Anal. Geom. 16 (1998) 419{444. MR1648844 30. B. Kostant, Holonomy and the lie algebra of infinitesimal motions of a Riemannian manifold, Trans. Amer. Math. Soc. 80 (1955) 528{542. MR0084825 31. D. Joyce, The hypercomplex quotient and the quaternionic quotient, Math. Ann. 290 (1991) 323{340. MR1109637 32. A. Gray, L. Hervella, The sixteen classes of almost Hermitian manifolds and their linear invariants, Ann. Mat. Pura Appl. 123 (1980) 35{58. MR0581924 33. M. Atiyah, R. Bott, The moment map and equivariant cohomology, Topology 23 (1984) 1{28. MR0721448 34. K. Galicki, A generalization of the momentum mapping, Comm. Math. Phys. 108 (1987) 117{138. MR0872143 35. S. Salamon, A tour of exceptional geometry, Milan J. Math. 71 (2003) 59{94. MR2120916 36. W. Fulton, J. Harris, Representation Theory: A First Course, in: GTM, vol. 129, Springer, 1991. MR1153249 37. A. Besse, Einstein Manifolds, in: Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 10, Springer-Verlag, 1987. MR0867684 38. F. Battaglia, S1 quotients of quaternion-K¨ahlermanifolds, Proc. Amer. Math. Soc. 124 (1996) 2185{2192. MR1307492 39. R. Miyaoka, The Bryant{Salamon G2-manifolds and hypersurface geometry, math- ph/0605074. 40. A. Gambioli, Eight-dimensional SU(3)-manifolds of cohomogeneity one, Ann. Global Anal. Geom. 34 (2008) 77{100. MR2415180 41. N. Hitchin, Stable forms and special metrics, in: Global Differential Geometry: The Mathematical Legacy of Alfred Gray, in: Contemp. Math., vol. 288, American Math. Soc., 2001, pp. 70{89. MR1871001 42. D. Conti, S. Salamon, Generalized Killing spinors in dimension 5, Trans. Amer. Math. Soc. 359 (2007) 5319{5343. MR2327032 43. V. Apostolov, S. Salamon, K¨ahlerreduction of metrics with holonomy G2, Comm. Math. Phys. 246 (2004) 43{61. MR2044890 44. S.M. Salamon, Almost parallel structures, in: Global Differential Geometry: The Mathematical Legacy of Alfred Gray, in: Contemp. Math., vol. 288, American Math. Soc., 2001, pp. 162{181. MR1871007 45. D. Conti, M. Fern´andez,Nilmanifolds with a calibrated G2-structure, Differential Geom. Appl. 29 (2011) 493{506. MR2811660 46. A. Fino, A. Tomassini, Generalized G2-manifolds and SU(3)-structures, Internat. J. Math. 19 (2008) 1147{1165. MR2466559 47. S. Salamon, Some reduced holonomy in dimensions 7 and 8, in: Workshop on Generalized Geometry and Flux Compactifications, DESY, Hamburg, 2007, http://www.desy.de/uni-th/stringth/ggfl/. Note: This list reflects references listed in the original paper as accurately as possible with no attempt to correct errors. c Copyright American Mathematical Society 2016 Citations From References: 5 From Reviews: 0 MR3132080 (Review) 53C25 53C29 53C44 53D20 Madsen, Thomas Bruun (4-LNDKC); Salamon, Simon [Salamon, Simon M.] (4-LNDKC) Half-flat structures on S3 × S3. (English summary) Ann. Global Anal. Geom. 44 (2013), no. 4, 369{390. A half-flat SU(3)-structure on a 6-dimensional manifold M is defined by a pair consisting of a 3-form and a 4-form which satisfy some compatibility conditions. From a half-flat SU(3)-structure one can reconstruct a metric g with holonomy group G2 via Hitchin flow. In the simplest case when M is a nearly-K¨ahlerspace, the metric g is the conical metric associated with M. The authors describe left-invariant half-flat SU(3)-structures on the Lie group S3 × S3 using the representation theory of the group SO(4) and matrix algebra. In particular, it is proven that on this group there exists unique left-invariant nearly-K¨ahlerstructure. The authors give a description of the moduli space of left-invariant half-flat SU(3)- structures in terms of matrix algebra. It is proven that essentially the moduli space is a finite-dimensional symplectic quotient. The matrix algebra is used also to simplify and interpret the Hitchin flow equations for the associated cohomogeneity one Ricci-flat metrics with holonomy G2. In the final part of the paper, the authors present results of a numerical study of Hitchin's evolution equations for S3 ×S3. They recover metrics that behave asymptotically locally conically. Dmitri˘ıVladimir Alekseevsky References 1. Apostolov, V., Salamon, S.: K¨ahlerreduction of metrics with holonomy G2. Comm. Math. Phys. 246(1), 43{61 (2004) MR2044890 2. Atiyah, M., Maldacena, J., Vafa, C.: An M-theory flop as a large N duality. Strings, branes, and M-theory. J. Math. Phys. 42(7), 3209{3220 (2001) MR1840340 3. Bar, C.: Real Killing spinors and holonomy. Comm. Math. Phys. 154(3), 509{521 (1993) MR1224089 4. Bedulli, L., Vezzoni, L,: The Ricci tensor of SU(3)-manifolds. J. Geom. Phys. 57(4), 1125{1146 (2007) MR2287296 5. Brandhuber, A.: G2 holonomy spaces; from invariant three-forms. Nuclear Phys. B 629(1{3), 393{416 (2002) MR1903163 6. Brandhuber, A., Gomis, J., Gubser, S., Gukov, S.: Gauge theory at large N and new G2 holonomy metrics. Nuclear Phys. B 611(1{3), 179{204 (2001) MR1857379 7. Bryant, R.: Non-embedding and non-extension results in special holonomy. The many facets of geometry. Oxford Cniversity Press, Oxford (2010) MR2681703 8. Bryant, R., Salamon, S.: On the construction of some complete metrics with excep- tional holonomy. Duke Math. J. 58(3), 829{850(1989) MR1016448 9. Butruille, J.-P.: Espacede twisteurs d'une vari´et´epresque hermitienne de dimension 6. Ann. Inst. Fourier (Grenoble) 57(5), 1451{485 (2007) MR2364136 10. Butruille J.-P.: Homogeneous nearly Kahler manifolds.
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