On Bismut flat manifolds

Wang, Qingsong; Yang, Bo; Zheng, Fangyang

doi:10.1090/tran/8083

On Bismut flat manifolds
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by Qingsong Wang, Bo Yang and Fangyang Zheng PDF

Trans. Amer. Math. Soc. 373 (2020), 5747-5772 Request permission

Abstract:

In this paper, we give a classification of all compact Hermitian manifolds with flat Bismut connection. We show that the torsion tensor of such a manifold must be parallel, thus the universal cover of such a manifold is a Lie group equipped with a bi-invariant metric and a compatible left invariant complex structure. In particular, isosceles Hopf surfaces are the only Bismut flat compact non-Kähler surfaces, while central Calabi-Eckmann threefolds are the only simply-connected compact Bismut flat threefolds.

References

Ilka Agricola and Thomas Friedrich, A note on flat metric connections with antisymmetric torsion, Differential Geom. Appl. 28 (2010), no. 4, 480–487. MR 2651537, DOI 10.1016/j.difgeo.2010.01.004
Bogdan Alexandrov and Stefan Ivanov, Vanishing theorems on Hermitian manifolds, Differential Geom. Appl. 14 (2001), no. 3, 251–265. MR 1836272, DOI 10.1016/S0926-2245(01)00044-4
Jean-Michel Bismut, A local index theorem for non-Kähler manifolds, Math. Ann. 284 (1989), no. 4, 681–699. MR 1006380, DOI 10.1007/BF01443359
William M. Boothby, Hermitian manifolds with zero curvature, Michigan Math. J. 5 (1958), 229–233. MR 103990
Lev Borisov, Simon Salamon, and Jeff Viaclovsky, Twistor geometry and warped product orthogonal complex structures, Duke Math. J. 156 (2011), no. 1, 125–166. MR 2746390, DOI 10.1215/00127094-2010-068
Eugenio Calabi and Beno Eckmann, A class of compact, complex manifolds which are not algebraic, Ann. of Math. (2) 58 (1953), 494–500. MR 57539, DOI 10.2307/1969750
E. Cartan and J.A. Schouten, On the geometry of the group manifold of simple and semisimple groups, Proc. Amsterdam 29 (1926), 803-815.
E. Cartan and J. A. Schouten, On Riemannian manifolds admitting an absolute parallelism, Proc. Amsterdam 29 (1926), 933-946.
Jixiang Fu, On non-Kähler Calabi-Yau threefolds with balanced metrics, Proceedings of the International Congress of Mathematicians. Volume II, Hindustan Book Agency, New Delhi, 2010, pp. 705–716. MR 2827815
Ji-Xiang Fu and Shing-Tung Yau, The theory of superstring with flux on non-Kähler manifolds and the complex Monge-Ampère equation, J. Differential Geom. 78 (2008), no. 3, 369–428. MR 2396248
Jixiang Fu, Jun Li, and Shing-Tung Yau, Balanced metrics on non-Kähler Calabi-Yau threefolds, J. Differential Geom. 90 (2012), no. 1, 81–129. MR 2891478
Jixiang Fu, Zhizhang Wang, and Damin Wu, Form-type Calabi-Yau equations, Math. Res. Lett. 17 (2010), no. 5, 887–903. MR 2727616, DOI 10.4310/MRL.2010.v17.n5.a7
Jixiang Fu, Zhizhang Wang, and Damin Wu, Semilinear equations, the $\gamma _k$ function, and generalized Gauduchon metrics, J. Eur. Math. Soc. (JEMS) 15 (2013), no. 2, 659–680. MR 3017048, DOI 10.4171/JEMS/370
Paul Gauduchon, La $1$-forme de torsion d’une variété hermitienne compacte, Math. Ann. 267 (1984), no. 4, 495–518 (French). MR 742896, DOI 10.1007/BF01455968
Alfred Gray, Curvature identities for Hermitian and almost Hermitian manifolds, Tohoku Math. J. (2) 28 (1976), no. 4, 601–612. MR 436054, DOI 10.2748/tmj/1178240746
Bo Guan, Second-order estimates and regularity for fully nonlinear elliptic equations on Riemannian manifolds, Duke Math. J. 163 (2014), no. 8, 1491–1524. MR 3284698, DOI 10.1215/00127094-2713591
Bo Guan and Qun Li, Complex Monge-Ampère equations and totally real submanifolds, Adv. Math. 225 (2010), no. 3, 1185–1223. MR 2673728, DOI 10.1016/j.aim.2010.03.019
Bo Guan and Qun Li, A Monge-Ampère type fully nonlinear equation on Hermitian manifolds, Discrete Contin. Dyn. Syst. Ser. B 17 (2012), no. 6, 1991–1999. MR 2924449, DOI 10.3934/dcdsb.2012.17.1991
Bo Guan and Wei Sun, On a class of fully nonlinear elliptic equations on Hermitian manifolds, Calc. Var. Partial Differential Equations 54 (2015), no. 1, 901–916. MR 3385185, DOI 10.1007/s00526-014-0810-1
Dominic Joyce, Compact hypercomplex and quaternionic manifolds, J. Differential Geom. 35 (1992), no. 3, 743–761. MR 1163458
Masahide Kato, Erratum to: “Topology of Hopf surfaces” [J. Math. Soc. Japan 27 (1975), 222–238; MR0402128 (53 #5949)], J. Math. Soc. Japan 41 (1989), no. 1, 173–174. MR 972171, DOI 10.2969/jmsj/04110173
Gabriel Khan, Bo Yang, and Fangyang Zheng, The set of all orthogonal complex structures on the flat 6-tori, Adv. Math. 319 (2017), 451–471. MR 3695880, DOI 10.1016/j.aim.2017.08.037
K. Kodaira, On the structure of compact complex analytic surfaces. I, Amer. J. Math. 86 (1964), 751–798. MR 187255, DOI 10.2307/2373157
Jun Li and Shing-Tung Yau, The existence of supersymmetric string theory with torsion, J. Differential Geom. 70 (2005), no. 1, 143–181. MR 2192064
Ke-Feng Liu and Xiao-Kui Yang, Geometry of Hermitian manifolds, Internat. J. Math. 23 (2012), no. 6, 1250055, 40. MR 2925476, DOI 10.1142/S0129167X12500553
Kefeng Liu and Xiaokui Yang, Ricci curvatures on Hermitian manifolds, Trans. Amer. Math. Soc. 369 (2017), no. 7, 5157–5196. MR 3632564, DOI 10.1090/tran/7000
Kefeng Liu and Xiaokui Yang, Hermitian harmonic maps and non-degenerate curvatures, Math. Res. Lett. 21 (2014), no. 4, 831–862. MR 3275649, DOI 10.4310/MRL.2014.v21.n4.a12
John Milnor, Curvatures of left invariant metrics on Lie groups, Advances in Math. 21 (1976), no. 3, 293–329. MR 425012, DOI 10.1016/S0001-8708(76)80002-3
Harsh V. Pittie, The Dolbeault-cohomology ring of a compact, even-dimensional Lie group, Proc. Indian Acad. Sci. Math. Sci. 98 (1988), no. 2-3, 117–152. MR 994129, DOI 10.1007/BF02863632
S. M. Salamon, Orthogonal complex structures, Differential geometry and applications (Brno, 1995) Masaryk Univ., Brno, 1996, pp. 103–117. MR 1406329
Simon Salamon and Jeff Viaclovsky, Orthogonal complex structures on domains in $\Bbb R^4$, Math. Ann. 343 (2009), no. 4, 853–899. MR 2471604, DOI 10.1007/s00208-008-0293-5
H. Samelson, A class of complex-analytic manifolds, Portugal. Math. 12 (1953), 129–132. MR 59287
Gábor Székelyhidi, Valentino Tosatti, and Ben Weinkove, Gauduchon metrics with prescribed volume form, Acta Math. 219 (2017), no. 1, 181–211. MR 3765661, DOI 10.4310/ACTA.2017.v219.n1.a6
Jeffrey Streets and Gang Tian, Hermitian curvature flow, J. Eur. Math. Soc. (JEMS) 13 (2011), no. 3, 601–634. MR 2781927, DOI 10.4171/JEMS/262
Jeffrey Streets and Gang Tian, A parabolic flow of pluriclosed metrics, Int. Math. Res. Not. IMRN 16 (2010), 3101–3133. MR 2673720, DOI 10.1093/imrn/rnp237
Jeffrey Streets and Gang Tian, Symplectic curvature flow, J. Reine Angew. Math. 696 (2014), 143–185. MR 3276165, DOI 10.1515/crelle-2012-0107
Jeffrey Streets and Gang Tian, Regularity results for pluriclosed flow, Geom. Topol. 17 (2013), no. 4, 2389–2429. MR 3110582, DOI 10.2140/gt.2013.17.2389
Andrew Strominger, Superstrings with torsion, Nuclear Phys. B 274 (1986), no. 2, 253–284. MR 851702, DOI 10.1016/0550-3213(86)90286-5
Valentino Tosatti, Non-Kähler Calabi-Yau manifolds, Analysis, complex geometry, and mathematical physics: in honor of Duong H. Phong, Contemp. Math., vol. 644, Amer. Math. Soc., Providence, RI, 2015, pp. 261–277. MR 3372471, DOI 10.1090/conm/644/12770
Valentino Tosatti and Ben Weinkove, The complex Monge-Ampère equation on compact Hermitian manifolds, J. Amer. Math. Soc. 23 (2010), no. 4, 1187–1195. MR 2669712, DOI 10.1090/S0894-0347-2010-00673-X
Li-Sheng Tseng and Shing-Tung Yau, Non-Kähler Calabi-Yau manifolds, String-Math 2011, Proc. Sympos. Pure Math., vol. 85, Amer. Math. Soc., Providence, RI, 2012, pp. 241–254. MR 2985333, DOI 10.1090/pspum/085/1381
Hsien-Chung Wang, Closed manifolds with homogeneous complex structure, Amer. J. Math. 76 (1954), 1–32. MR 66011, DOI 10.2307/2372397
Joseph A. Wolf, On the geometry and classification of absolute parallelisms. I, J. Differential Geometry 6 (1971/72), 317–342. MR 312442
Joseph A. Wolf, On the geometry and classification of absolute parallelisms. II, J. Differential Geometry 7 (1972), 19–44. MR 312443
Bo Yang and Fangyang Zheng, On curvature tensors of Hermitian manifolds, Comm. Anal. Geom. 26 (2018), no. 5, 1195–1222. MR 3900484, DOI 10.4310/CAG.2018.v26.n5.a7
Kentaro Yano, Differential geometry on complex and almost complex spaces, International Series of Monographs in Pure and Applied Mathematics, Vol. 49, The Macmillan Company, New York, 1965. A Pergamon Press Book. MR 0187181
Fangyang Zheng, Complex differential geometry, AMS/IP Studies in Advanced Mathematics, vol. 18, American Mathematical Society, Providence, RI; International Press, Boston, MA, 2000. MR 1777835, DOI 10.1090/amsip/018