The intrinsic torsion of almost quaternion-Hermitian manifolds
Annales de l'Institut Fourier, Volume 58 (2008) no. 5, p. 1455-1497

We study the intrinsic torsion of almost quaternion-Hermitian manifolds via the exterior algebra. In particular, we show how it is determined by particular three-forms formed from simple combinations of the exterior derivatives of the local Kähler forms. This gives a practical method to compute the intrinsic torsion and is applied in a number of examples. In addition we find simple characterisations of HKT and QKT geometries entirely in the exterior algebra and compute how the intrinsic torsion changes under a twist construction.

Nous étudions la torsion intrinsèque des variétés presque hermitiennes quaternioniennes via l’algèbre extérieur. En particulier, nous montrons comment elle est déterminée par trois-formes particulières, formées à partir de simples combinaisons des différentielles extérieures des formes kählériennes locales. Ceci donne une méthode pratique pour calculer la torsion intrinsèque qui s’applique dans de nombreux exemples. En plus, nous trouvons des caractérisations simples des géométries HKT et QKT en utilisant l’algèbre extérieur et nous calculons la modification de la torsion intrinsèque pour une construction twistée.

DOI : https://doi.org/10.5802/aif.2390
Classification:  53C15,  53C10,  53C26,  53C80
Keywords: Almost Hermitian structure, almost quaternion-Hermitian structure, G-structure, intrinsic torsion, G-connection, HKT-manifold, QKT-manifold
@article{AIF_2008__58_5_1455_0,
     author = {Mart\'\i n Cabrera, Francisco and Swann, Andrew},
     title = {The intrinsic torsion of almost quaternion-Hermitian manifolds},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {58},
     number = {5},
     year = {2008},
     pages = {1455-1497},
     doi = {10.5802/aif.2390},
     mrnumber = {2445825},
     zbl = {1145.53017},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2008__58_5_1455_0}
}
Martín Cabrera, Francisco; Swann, Andrew. The intrinsic torsion of almost quaternion-Hermitian manifolds. Annales de l'Institut Fourier, Volume 58 (2008) no. 5, pp. 1455-1497. doi : 10.5802/aif.2390. http://www.numdam.org/item/AIF_2008__58_5_1455_0/

[1] Auslander, L.; Green, L.; Hahn, F. Flows on homogeneous spaces, Princeton University Press, Princeton, N.J., With the assistance of L. Markus and W. Massey, and an appendix by L. Greenberg. Annals of Mathematics Studies, No. 53 (1963) | Zbl 0106.36802

[2] Berger, Marcel Sur les groupes d’holonomie homogène des variétés à connexion affine et des variétés riemanniennes, Bull. Soc. Math. France, Tome 83 (1955), pp. 279-330 | Numdam | Zbl 0068.36002

[3] Bröcker, Theodor; Tom Dieck, Tammo Representations of compact Lie groups, Springer-Verlag, New York, Graduate Texts in Mathematics, Tome 98 (1985) | MR 781344 | Zbl 0581.22009

[4] Cleyton, Richard; Swann, Andrew Einstein metrics via intrinsic or parallel torsion, Math. Z., Tome 247 (2004) no. 3, pp. 513-528 | Article | MR 2114426 | Zbl 1069.53041

[5] Cordero, Luis A.; Fernández, Marisa; Gray, Alfred Minimal models in differential geometry, Proceedings of the Workshop on Recent Topics in Differential Geometry (Puerto de la Cruz, 1990), Univ. La Laguna, La Laguna (Informes) Tome 32 (1991), pp. 31-41 | MR 1127451 | Zbl 0736.53037

[6] Cordero, Luis A.; Fernández, Marisa; De León, Manuel On the quaternionic Heisenberg group, Boll. Un. Mat. Ital. A (7), Tome 1 (1987) no. 1, pp. 31-37 | MR 880099 | Zbl 0613.53016

[7] Cordero, Luis A.; Fernández, Marisa; De León, Manuel; Saralegui, Martín Compact symplectic four solvmanifolds without polarizations, Ann. Fac. Sci. Toulouse Math. (5), Tome 10 (1989) no. 2, pp. 193-198 | Article | Numdam | MR 1425485 | Zbl 0659.53032

[8] Fernández, M.; Moreiras, B. R. Symmetry properties of the covariant derivative of the fundamental 4-form of a quaternionic manifold, Riv. Mat. Univ. Parma (4), Tome 12 (1986), p. 249-256 (1987) | MR 913047 | Zbl 0631.53029

[9] Fernández, Marisa; Gray, Alfred Compact symplectic solvmanifolds not admitting complex structures, Geom. Dedicata, Tome 34 (1990) no. 3, pp. 295-299 | Article | MR 1066580 | Zbl 0703.53030

[10] Gates, S. J. Jr.; Hull, C. M.; Roček, M. Twisted multiplets and new supersymmetric nonlinear σ-models, Nuclear Phys. B, Tome 248 (1984) no. 1, pp. 157-186 | Article | MR 776369

[11] Grantcharov, Gueo; Poon, Yat Sun Geometry of hyper-Kähler connections with torsion, Comm. Math. Phys., Tome 213 (2000) no. 1, pp. 19-37 | Article | MR 1782143 | Zbl 0993.53016

[12] Gray, Alfred Minimal varieties and almost Hermitian submanifolds, Michigan Math. J., Tome 12 (1965), pp. 273-287 | Article | MR 184185 | Zbl 0132.16702

[13] Gray, Alfred; Hervella, Luis M. The sixteen classes of almost Hermitian manifolds and their linear invariants, Ann. Mat. Pura Appl. (4), Tome 123 (1980), pp. 35-58 | Article | MR 581924 | Zbl 0444.53032

[14] Hitchin, N. J. The self-duality equations on a Riemann surface, Proc. London Math. Soc. (3), Tome 55 (1987) no. 1, pp. 59-126 | Article | MR 887284 | Zbl 0634.53045

[15] Howe, P. S.; Opfermann, A.; Papadopoulos, G. Twistor spaces for QKT manifolds, Comm. Math. Phys., Tome 197 (1998) no. 3, pp. 713-727 | Article | MR 1652783 | Zbl 0941.53031

[16] Howe, P. S.; Papadopoulos, G. Further remarks on the geometry of two-dimensional nonlinear σ models, Classical Quantum Gravity, Tome 5 (1988) no. 12, pp. 1647-1661 | Article | MR 973266 | Zbl 0654.53071

[17] Howe, P. S.; Papadopoulos, G. Twistor spaces for hyper-Kähler manifolds with torsion, Phys. Lett. B, Tome 379 (1996) no. 1-4, pp. 80-86 | Article | MR 1396267

[18] Ivanov, Stefan Geometry of quaternionic Kähler connections with torsion, J. Geom. Phys., Tome 41 (2002) no. 3, pp. 235-257 | Article | MR 1877929 | Zbl 1007.53054

[19] Joyce, Dominic Compact hypercomplex and quaternionic manifolds, J. Differential Geom., Tome 35 (1992) no. 3, pp. 743-761 | MR 1163458 | Zbl 0735.53050

[20] Martín Cabrera, Francisco Almost quaternion-Hermitian manifolds, Ann. Global Anal. Geom., Tome 25 (2004) no. 3, pp. 277-301 | Article | MR 2053763 | Zbl 1061.53030

[21] Martín Cabrera, Francisco; Swann, Andrew Almost Hermitian structures and quaternionic geometries, Differential Geom. Appl., Tome 21 (2004) no. 2, pp. 199-214 | Article | MR 2073825 | Zbl 1062.53034

[22] Obata, Morio Affine connections on manifolds with almost complex, quaternion or Hermitian structure, Jap. J. Math., Tome 26 (1956), pp. 43-77 | MR 95290 | Zbl 0089.17203

[23] Salamon, S. M. Quaternionic Kähler manifolds, Invent. Math., Tome 67 (1982) no. 1, pp. 143-171 | Article | MR 664330 | Zbl 0486.53048

[24] Salamon, S. M. Differential geometry of quaternionic manifolds, Ann. Sci. École Norm. Sup. (4), Tome 19 (1986) no. 1, pp. 31-55 | Numdam | MR 860810 | Zbl 0616.53023

[25] Salamon, S. M. Riemannian geometry and holonomy groups, Longman Scientific & Technical, Harlow, Pitman Research Notes in Mathematics Series, Tome 201 (1989) | Zbl 0685.53001

[26] Salamon, S. M. Almost parallel structures, Global differential geometry: the mathematical legacy of Alfred Gray (Bilbao, 2000), Amer. Math. Soc., Providence, RI (Contemp. Math.) Tome 288 (2001), pp. 162-181 | MR 1871007 | Zbl 1008.53043

[27] Swann, Andrew Aspects symplectiques de la géométrie quaternionique, C. R. Acad. Sci. Paris Sér. I Math., Tome 308 (1989) no. 7, pp. 225-228 | MR 986384 | Zbl 0661.53023

[28] Swann, Andrew T is for twist, Proceedings of the XV International Workshop on Geometry and Physics (Puerto de la Cruz, 2006), Spanish Royal Mathematical Society, Madrid, Tome 10 (2007), pp. 83-94