Abstract
In the first part, we define and investigate new classes of almost 3-contact metric manifolds, with two guiding ideas in mind: first, what geometric objects are best suited for capturing the key properties of almost 3-contact metric manifolds, and second, the new classes should admit ‘good’ metric connections with skew torsion. In particular, we introduce the Reeb commutator function and the Reeb Killing function, we define the new classes of canonical almost 3-contact metric manifolds and of 3-(α, δ)-Sasaki manifolds (including as special cases 3-Sasaki manifolds, quaternionic Heisenberg groups, and many others) and prove that the latter are hypernormal, thus generalizing a seminal result of Kashiwada. We study their behaviour under a new class of deformations, called 𝓗-homothetic deformations, and prove that they admit an underlying quaternionic contact structure, from which we deduce the Ricci curvature. For example, a 3-(α, δ)-Sasaki manifold is Einstein either if α = δ (the 3-α-Sasaki case) or if δ = (2n + 3)α, where dim M = 4n + 3.
In the second part we find these adapted connections. We start with a very general notion of φ-compatible connections, where φ denotes any element of the associated sphere of almost contact structures, and make them unique by a certain extra condition, thus yielding the notion of canonical connection (they exist exactly on canonical manifolds, hence the name). For 3-(α, δ)-Sasaki manifolds, we compute the torsion of this connection explicitly and we prove that it is parallel, we describe the holonomy, the ∇-Ricci curvature, and we show that the metric cone is a HKT-manifold. In dimension 7, we construct a cocalibrated G2-structure inducing the canonical connection and we prove the existence of four generalized Killing spinors.
Communicated by: F. Duzaar
Acknowledgements
G. Dileo acknowledges the financial support of DAAD for a research stay at Philipps-Universität Marburg in the period April–June 2016, under the Programme Research Stays for University Academics and Scientists. She thanks Philipps-Universität for its kind hospitality.
References
[1] I. Agricola, The Srní lectures on non-integrable geometries with torsion. Arch. Math. (Brno)42 (2006), 5–84. MR2322400 Zbl 1164.53300Search in Google Scholar
[2] I. Agricola, J. Becker-Bender, H. Kim, Twistorial eigenvalue estimates for generalized Dirac operators with torsion. Adv. Math.243 (2013), 296–329. MR3062748 Zbl 1287.5801510.1016/j.aim.2013.05.001Search in Google Scholar
[3] I. Agricola, S. G. Chiossi, T. Friedrich, J. Höll, Spinorial description of SU(3)- and G2-manifolds. J. Geom. Phys.98 (2015), 535–555. MR3414976 Zbl 1333.5303710.1016/j.geomphys.2015.08.023Search in Google Scholar
[4] I. Agricola, A. C. Ferreira, Einstein manifolds with skew torsion. Q. J. Math.65 (2014), 717–741. MR3261964 Zbl 1368.5303410.1093/qmath/hat050Search in Google Scholar
[5] I. Agricola, A. C. Ferreira, R. Storm, Quaternionic Heisenberg groups as naturally reductive homogeneous spaces. Int. J. Geom. Methods Mod. Phys.12 (2015), 1560007, 10 pages. MR3400648 Zbl 1333.5306910.1142/S0219887815600075Search in Google Scholar
[6] I. Agricola, T. Friedrich, 3-Sasakian manifolds in dimension seven, their spinors and G2-structures. J. Geom. Phys.60 (2010), 326–332. MR2587396 Zbl 1188.5304110.1016/j.geomphys.2009.10.003Search in Google Scholar
[7] I. Agricola, T. Friedrich, A note on flat metric connections with antisymmetric torsion. Differential Geom. Appl.28 (2010), 480–487. MR2651537 Zbl 1201.5305210.1016/j.difgeo.2010.01.004Search in Google Scholar
[8] I. Agricola, J. Höll, Cones of G manifolds and Killing spinors with skew torsion. Ann. Mat. Pura Appl. (4)194 (2015), 673–718. MR3345660 Zbl 1319.5304310.1007/s10231-013-0393-zSearch in Google Scholar
[9] H. Baum, T. Friedrich, R. Grunewald, I. Kath, Twistors and Killing spinors on Riemannian manifolds, volume 124 of Teubner-Texte zur Mathematik. B. G. Teubner Verlagsgesellschaft, Stuttgart 1991. MR1164864 Zbl 0734.53003Search in Google Scholar
[10] O. Biquard, Métriques ďEinstein asymptotiquement symétriques. Astérisque no. 265 (2000), vi+109. MR1760319 Zbl 0967.53030Search in Google Scholar
[11] D. E. Blair, Riemannian geometry of contact and symplectic manifolds. Birkhäuser 2010. MR2682326 Zbl 1246.5300110.1007/978-0-8176-4959-3Search in Google Scholar
[12] C. P. Boyer, K. Galicki, Sasakian geometry. Oxford Univ. Press 2008. MR2382957 Zbl 1155.5300210.1093/acprof:oso/9780198564959.001.0001Search in Google Scholar
[13] B. Cappelletti Montano, 3-structures with torsion. Differential Geom. Appl.27 (2009), 496–506. MR2547828 Zbl 1176.5304110.1016/j.difgeo.2009.01.009Search in Google Scholar
[14] B. Cappelletti Montano, A. De Nicola, 3-Sasakian manifolds, 3-cosymplectic manifolds and Darboux theorem. J. Geom. Phys.57 (2007), 2509–2520. MR2369836 Zbl 1193.5309210.1016/j.geomphys.2007.09.001Search in Google Scholar
[15] B. Cappelletti Montano, A. De Nicola, G. Dileo, 3-quasi-Sasakian manifolds. Ann. Global Anal. Geom.33 (2008), 397–409. MR2395194 Zbl 1186.5304110.1007/s10455-007-9093-5Search in Google Scholar
[16] B. Cappelletti Montano, A. De Nicola, G. Dileo, The geometry of 3-quasi-Sasakian manifolds. Internat. J. Math.20 (2009), 1081–1105. MR2572593 Zbl 1187.5302610.1142/S0129167X09005662Search in Google Scholar
[17] B. Cappelletti-Montano, A. De Nicola, I. Yudin, Hard Lefschetz theorem for Sasakian manifolds. J. Differential Geom.101 (2015), 47–66. MR3356069 Zbl 1322.5800210.4310/jdg/1433975483Search in Google Scholar
[18] B. Cappelletti-Montano, A. De Nicola, I. Yudin, Cosymplectic p-spheres. J. Geom. Phys.100 (2016), 68–79.10.1016/j.geomphys.2015.11.005Search in Google Scholar
[19] E. Cartan, J. A. Schouten, On Riemannian manifolds admitting an absolute parallelism. Proc. Amsterdam29 (1926), 933–946. JFM 52.0744.02Search in Google Scholar
[20] I. Chrysikos, Invariant connections with skew-torsion and ∇-Einstein manifolds. J. Lie Theory26 (2016), 11–48. MR3384980 Zbl 1342.53072Search in Google Scholar
[21] D. Conti, T. B. Madsen, The odd side of torsion geometry. Ann. Mat. Pura Appl. (4)193 (2014), 1041–1067. MR3237915 Zbl 1314.5305510.1007/s10231-012-0314-6Search in Google Scholar
[22] D. Conti, S. Salamon, Reduced holonomy, hypersurfaces and extensions. Int. J. Geom. Methods Mod. Phys.3 (2006), 899–912. MR2264396 Zbl 1116.5303110.1142/S021988780600148XSearch in Google Scholar
[23] J. E. D’Atri, H. K. Nickerson, The existence of special orthonormal frames. J. Differential Geometry2 (1968), 393–409. MR0248682 Zbl 0179.5060110.4310/jdg/1214428656Search in Google Scholar
[24] G. Dileo, A. Lotta, Riemannian almost CR manifolds with torsion. Illinois J. Math.58 (2014), 807–846. MR3395964 Zbl 1328.5303610.1215/ijm/1441790391Search in Google Scholar
[25] C. Draper, A. Garví n, F. J. Palomo, Invariant affine connections on odd-dimensional spheres. Ann. Global Anal. Geom.49 (2016), 213–251. MR3485984 Zbl 1342.5307310.1007/s10455-015-9489-6Search in Google Scholar
[26] D. Duchemin, Quaternionic contact structures in dimension 7. Ann. Inst. Fourier (Grenoble)56 (2006), 851–885. MR2266881 Zbl 1122.5302510.5802/aif.2203Search in Google Scholar
[27] M. Falcitelli, S. Ianus, A. M. Pastore, Riemannian submersions and related topics. World Scientific Publishing Co., River Edge, NJ 2004. MR2110043 Zbl 1067.5301610.1142/9789812562333Search in Google Scholar
[28] T. Friedrich, G2-manifolds with parallel characteristic torsion. Differential Geom. Appl.25 (2007), 632–648. MR2373939 Zbl 1141.5301910.1016/j.difgeo.2007.06.010Search in Google Scholar
[29] T. Friedrich, S. Ivanov, Parallel spinors and connections with skew-symmetric torsion in string theory. Asian J. Math.6 (2002), 303–335. MR1928632 Zbl 1127.5330410.4310/AJM.2002.v6.n2.a5Search in Google Scholar
[30] T. Friedrich, I. Kath, 7-dimensional compact Riemannian manifolds with Killing spinors. Comm. Math. Phys.133 (1990), 543–561. MR1079795 Zbl 0722.5303810.1007/BF02097009Search in Google Scholar
[31] T. Friedrich, I. Kath, A. Moroianu, U. Semmelmann, On nearly parallel G2-structures. J. Geom. Phys.23 (1997), 259–286. MR1484591 Zbl 0898.5303810.1016/S0393-0440(97)80004-6Search in Google Scholar
[32] K. Galicki, S. Salamon, Betti numbers of 3-Sasakian manifolds. Geom. Dedicata63 (1996), 45–68. MR1413621 Zbl 0859.5303110.1007/BF00181185Search in Google Scholar
[33] G. Grantcharov, Y. S. Poon, Geometry of hyper-Kähler connections with torsion. Comm. Math. Phys.213 (2000), 19–37. MR1782143 Zbl 0993.5301610.1007/s002200000231Search in Google Scholar
[34] T. Houri, H. Takeuchi, Y. Yasui, A deformation of Sasakian structure in the presence of torsion and supergravity solutions. Classical Quantum Gravity30 (2013), 135008, 31 pages. MR3072914 Zbl 1273.8314910.1088/0264-9381/30/13/135008Search in Google Scholar
[35] S. Ishihara, Quaternion Kählerian manifolds. J. Differential Geometry9 (1974), 483–500. MR0348687 Zbl 0297.5301410.4310/jdg/1214432544Search in Google Scholar
[36] S. Ivanov, I. Minchev, D. Vassilev, Quaternionic contact Einstein structures and the quaternionic contact Yamabe problem. Mem. Amer. Math. Soc.231 (2014), no. 1086, vi+82 pages. MR3235632 Zbl 1307.53039Search in Google Scholar
[37] S. Ivanov, I. Minchev, D. Vassilev, Quaternionic contact Einstein manifolds. Math. Res. Lett.23 (2016), 1405–1432. MR3601072 Zbl 1372.5304910.4310/MRL.2016.v23.n5.a8Search in Google Scholar
[38] T. Kashiwada, A note on Hitchin’s lemma. Tensor (N.S.)60 (1998), 323–326. MR1835606 Zbl 1040.53060Search in Google Scholar
[39] T. Kashiwada, On a contact 3-structure. Math. Z.238 (2001), 829–832. MR1872576 Zbl 1004.5305810.1007/s002090100279Search in Google Scholar
[40] M. Konishi, On manifolds with Sasakian 3-structure over quaternion Kaehler manifolds. Kōdai Math. Sem. Rep.26 (1974/75), 194–200. MR0377782 Zbl 0308.5303510.2996/kmj/1138847001Search in Google Scholar
[41] Y.-y. Kuo, On almost contact 3-structure. Tohoku Math. J. (2)22 (1970), 325–332. MR0278225 Zbl 0205.2580110.2748/tmj/1178242759Search in Google Scholar
[42] C. Puhle, Almost contact metric 5-manifolds and connections with torsion. Differential Geom. Appl.30 (2012), 85–106. MR2871707 Zbl 1236.5306510.1016/j.difgeo.2011.11.007Search in Google Scholar
[43] C. Puhle, On generalized quasi-Sasaki manifolds. Differential Geom. Appl.31 (2013), 217–229. MR3032644 Zbl 1281.5303610.1016/j.difgeo.2013.01.003Search in Google Scholar
[44] S. M. Salamon, Quaternion-Kähler geometry. In: Surveys in differential geometry: essays on Einstein manifolds, volume 6 of Surv. Differ. Geom., 83–121, Int. Press, Boston, MA 1999. MR1798608 Zbl 1003.5303910.4310/SDG.2001.v6.n1.a5Search in Google Scholar
[45] S. Tanno, Remarks on a triple of K-contact structures. Tohoku Math. J. 248 (1996), 519–531. MR1419082 Zbl 0881.5304010.2748/tmj/1178225296Search in Google Scholar
[46] L. Vanhecke, D. Janssens, Almost contact structures and curvature tensors. Kodai Math. J.4 (1981), 1–27. MR615665 Zbl 0472.5304310.2996/kmj/1138036310Search in Google Scholar
[47] J. A. Wolf, On the geometry and classification of absolute parallelisms. I. J. Differential Geometry6 (1971/72), 317–342. MR0312442 Zbl 0251.5301410.4310/jdg/1214430496Search in Google Scholar
[48] J. A. Wolf, On the geometry and classification of absolute parallelisms. II. J. Differential Geometry7 (1972), 19–44. MR0312443 Zbl 0276.5301710.4310/jdg/1214430818Search in Google Scholar
[49] K. Yano, S. Ishihara, M. Konishi, Normality of almost contact 3-structure. Tohoku Math. J. (2) 25 (1973), 167–175. MR0336644 Zbl 0268.5301610.2748/tmj/1178241375Search in Google Scholar
© 2020 Walter de Gruyter GmbH, Berlin/Boston
