The intrinsic torsion of almost quaternion-Hermitian manifolds  [ La torsion intrinsèque des variétés presque hermitiennes quaternioniennes ]
Annales de l'Institut Fourier, Tome 58 (2008) no. 5, pp. 1455-1497.

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.

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.

DOI : https://doi.org/10.5802/aif.2390
Classification : 53C15,  53C10,  53C26,  53C80
Mots clés : structure presque hermitienne, structure presque hermitienne quaternionienne, G-structure, torsion intrinsèque, G-connexion, HKT-variété, QKT-variété
@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},
     pages = {1455--1497},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {58},
     number = {5},
     year = {2008},
     doi = {10.5802/aif.2390},
     mrnumber = {2445825},
     zbl = {1145.53017},
     language = {en},
     url = {archive.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, Tome 58 (2008) no. 5, pp. 1455-1497. doi : 10.5802/aif.2390. http://archive.numdam.org/item/AIF_2008__58_5_1455_0/

[1] Auslander, L.; Green, L.; Hahn, F. Flows on homogeneous spaces, With the assistance of L. Markus and W. Massey, and an appendix by L. Greenberg. Annals of Mathematics Studies, No. 53, Princeton University Press, Princeton, N.J., 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, Volume 83 (1955), pp. 279-330 | Numdam | Zbl 0068.36002

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

[4] Cleyton, Richard; Swann, Andrew Einstein metrics via intrinsic or parallel torsion, Math. Z., Volume 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) (Informes) Volume 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), Volume 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), Volume 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), Volume 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, Volume 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, Volume 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., Volume 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., Volume 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), Volume 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), Volume 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., Volume 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, Volume 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, Volume 379 (1996) no. 1-4, pp. 80-86 | Article | MR 1396267

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

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

[20] Martín Cabrera, Francisco Almost quaternion-Hermitian manifolds, Ann. Global Anal. Geom., Volume 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., Volume 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., Volume 26 (1956), pp. 43-77 | MR 95290 | Zbl 0089.17203

[23] Salamon, S. M. Quaternionic Kähler manifolds, Invent. Math., Volume 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), Volume 19 (1986) no. 1, pp. 31-55 | Numdam | MR 860810 | Zbl 0616.53023

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

[26] Salamon, S. M. Almost parallel structures, Global differential geometry: the mathematical legacy of Alfred Gray (Bilbao, 2000) (Contemp. Math.) Volume 288, Amer. Math. Soc., Providence, RI, 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., Volume 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), Volume 10 (2007), pp. 83-94