×

Almost Hermitian 6-manifolds revisited. (English) Zbl 1075.53036

Let \((M,g)\) be a Riemannian manifold equipped with a \(G\)-structure. A connection is called a characteristic connection \(\nabla^c\) of the \(G\)-structure if it is a \(G\)-connection with totally skew-symmetric torsion \(T^c\), i.e., \(T^c\) corresponds to a 3-form. For a Riemannian naturally reductive space \(M = G_1/G\) the characteristic connection of the corresponding \(G\)-reduction coincides with the canonical connection of the reductive space and moreover \(T^c\) and the curvature tensor \(R^c\) are parallel. This parallelism does not hold in general but \(\nabla^cR^c = 0\) is an important condition.
In this paper the authors study the case where the \(G\)-structure is given by an almost Hermitian structure \(J\) and they focus on the 4- and 6-dimensional case. For these cases they derive conditions for the existence of characteristic connections and then study these cases by using a decomposition of the space of 3-forms. For \(\dim M = 4\) they obtain that \((M^4,g,J)\) admits a characteristic connection if and only if it is a Hermitian manifold and moreover, \(T^c\) is parallel if and only if \((M^4,g,J)\) is a generalized Hopf manifold (i.e., the Lee form \(\delta \Omega \circ J\) is parallel with respect to the Levi Civita connection, \(\Omega\) being the Kähler form).
Next, they consider the 6-dimensional spaces where they discuss their decompositions and combine them with the known one leading to the well-known classification of almost Hermitian structures into 16 types. Then they specialize to the special classes of nearly Kähler spaces and of \(G_1\)-spaces, respectively. For the first class they indicate a proof of a result of Kirichenko stating that the characteristic connection exists and has parallel torsion \(T^c\). For the second case they prove that if \((M,g,J)\) is of type \(G_1\) (a class containing the naturally reductive spaces) then it admits a unique characteristic connection. Furthermore, they study the case where its torsion \(T^c\) is parallel and prove that there are only two types of semi-Kähler spaces: the twistor space of a 4-dimensional self-dual Einstein space and the invariant Hermitian structure on SL\((2,\mathbb C)\), respectively. Finally, all naturally reductive semi-Kähler spaces with small isotropy group of \(\nabla^c\) are determined.

MSC:

53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] Abbena, E.; Garbiero, S.; Salamon, S., Almost Hermitian geometry on six-dimensional nilmanifolds, Ann. sci. norm. sup, 30, 147-170, (2001) · Zbl 1072.53010
[2] J.F. Adams, Lectures on Lie Groups, University of Chicago Press, 1969. · Zbl 0206.31604
[3] Agricola, I., Connections on naturally reductive spaces, their Dirac operator and homogeneous models in string theory, Comm. math. phys, 232, 535-563, (2003) · Zbl 1032.53041
[4] Agricola, I.; Friedrich, T., On the holonomy of connections with skew-symmetric torsion, Math. ann, 328, 711-748, (2004) · Zbl 1055.53031
[5] Agricola, I.; Friedrich, T., The Casimir operator of a metric connection with skew-symmetric torsion, J. geom. phys, 50, 188-204, (2004) · Zbl 1080.53043
[6] B. Alexandrov, Sp(n)U(1)-connections with parallel totally skew-symmetric torsion, math.dg/0311248. · Zbl 1107.53012
[7] J.E. D’Atri, W. Ziller, Naturally reductive metrics and Einstein metrics on compact Lie groups, Memoirs of AMS No. 215, 1979. · Zbl 0404.53044
[8] H. Baum, T. Friedrich, R. Grunewald, I. Kath, Twistors and Killing spinors on Riemannian manifolds, Teubner-Texte zur Mathematik No. 124, Teubner-Verlag Leipzig, Stuttgart, 1991. · Zbl 0734.53003
[9] Belgun, F., On the metric structure of non-Kähler complex surfaces, Math. ann, 317, 1-40, (2000) · Zbl 0988.32017
[10] Belgun, F.; Moroianu, A., Nearly Kähler 6-manifolds with reduced holonomy, Ann. global anal. geom, 19, 307-319, (2001) · Zbl 0992.53037
[11] Bismut, J.M., A local index theorem for non-Kählerian manifolds, Math. ann, 284, 681-699, (1989) · Zbl 0666.58042
[12] J.-B. Butruille, Classification des varietes approximativement kähleriennes homogenes, math.dg/0401152.
[13] S. Chiossi, S. Salamon, The intrinsic torsion of SU(3) and G_{2}-structures, Differential Geometry, Valencia, 2001, Word Scientific, River Edge, NJ, 2002, pp. 115-133. · Zbl 1024.53018
[14] R. Cleyton, A. Swann, Einstein metrics via intrinsic or parallel torsion, math.dg/0211446. · Zbl 1069.53041
[15] Falcitelli, M.; Farinola, A.; Salamon, S., Almost-Hermitian geometry, Diff. geom. appl, 4, 259-282, (1994) · Zbl 0813.53044
[16] A. Fino, M. Porton, S. Salamon, Families of strong KT structures in six dimensions, math.dg/0209259.
[17] Friedrich, T., Der erste eigenwert des Dirac operators einer kompakten riemannschen mannigfaltigkeit nichtnegativer skalarkrümmung, Math. nachr, 97, 117-146, (1980) · Zbl 0462.53027
[18] T. Friedrich, Dirac Operators in Riemannian Geometry, Graduate Studies in Mathematics, vol. 25, AMS, Providence, 2000. · Zbl 0949.58032
[19] Friedrich, T., On types of non-integrable geometries, Rend. circ. mat. di Palermo, 71, 99-113, (2003) · Zbl 1079.53041
[20] Friedrich, T., Spin(9)-structures and connections with totally skew-symmetric torsion, J. geom. phys, 47, 197-206, (2003) · Zbl 1039.53049
[21] Friedrich, T.; Grunewald, R., On the first eigenvalue of the Dirac operator on six-dimensional manifolds, Ann. global anal. geom, 3, 265-273, (1985) · Zbl 0577.58034
[22] Friedrich, T.; Ivanov, S., Parallel spinors and connections with skew-symmetric torsion in string theory, Asian J. math, 6, 303-336, (2002) · Zbl 1127.53304
[23] Friedrich, T.; Kurke, H., Compact four-dimensional self-dual Einstein manifolds with positive scalar curvature, Math. nachr, 106, 271-299, (1982) · Zbl 0503.53035
[24] D. Grantcharov, G. Grantcharov, Y.S. Poon, Calabi-Yau connections with torsion on toric bundles, math.dg/0306207. · Zbl 1171.53044
[25] Gray, A., Almost complex submanifolds of the six sphere, Proc. am. math. soc, 20, 277-279, (1969) · Zbl 0165.55803
[26] Gray, A., Six-dimensional almost complex manifolds defined by means of the three-fold vector cross products, Tohoku math. J. II ser, 21, 614-620, (1969) · Zbl 0192.59002
[27] Gray, A., Nearly Kähler manifolds, J. diff. geom, 4, 283-310, (1970) · Zbl 0201.54401
[28] Gray, A., The structure of nearly Kähler manifolds, Math. ann, 223, 233-248, (1976) · Zbl 0345.53019
[29] Gray, A.; Hervella, L., The sixteen classes of almost Hermitian manifolds and their linear invariants, Ann. di mat. pura ed appl, 123, 35-58, (1980) · Zbl 0444.53032
[30] Grunewald, R., Six-dimensional Riemannian manifolds with real Killing spinors, Ann. glob. anal. geom, 8, 43-59, (1990) · Zbl 0704.53050
[31] Hitchin, N., On compact four-dimensional Einstein manifolds, J. diff. geom, 9, 435-442, (1974) · Zbl 0281.53039
[32] Hitchin, N., Kählerian twistor spaces, Proc. lond. math. soc. III ser, 43, 133-150, (1981) · Zbl 0474.14024
[33] Kirichenko, V., K-spaces of maximal rank, Mat. zam, 22, 465-476, (1977)
[34] Kostant, B., A cubic Dirac operator and the emergence of Euler number multiplets of representations for equal rank subgroups, Duke math. J, 100, 447-501, (1999) · Zbl 0952.17005
[35] Matsumoto, M., On 6-dimensional almost tachibana spaces, Tensor N. S, 23, 250-282, (1972) · Zbl 0236.53048
[36] Nagy, P.-A., Nearly Kähler geometry and Riemannian foliations, Asian J. math, 6, 481-504, (2002) · Zbl 1041.53021
[37] S. Sternberg, Lie algebras, preprint, November 1999.
[38] Takamatsu, K., Some properties of 6-dimensional K-spaces, Kodai math. sem. rep, 23, 215-232, (1971) · Zbl 0222.53036
[39] Vaisman, I., Locally conformal Kähler manifolds with parallel Lee form, Rendiconti di matem. roma, 12, 263-284, (1979) · Zbl 0447.53032
[40] K. Yano, M. Kon, Structures on Manifolds, World Scientific, 1984. · Zbl 0557.53001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.