Abstract
The Chern connection was defined in 1946 in the almost Hermitian geometry, and in 2005 in the almost paraHermitian context. In 2017, it was proved that the condition defining the Chern connection does not determine a unique connection in the other two \((J^2 = \pm 1)\)-metric geometries (almost Norden and almost product Riemannian with null trace). Now, we prove that there exists a unique canonical connection satisfying the Chern condition in these two geometries if and only if the manifold is of type \({\mathcal {G}}_1\). For such a characterization, we need an exhaustive study of the plane of quasi-canonical connections, defined by the Levi-Civita connection and the line of canonical connections. Connections with totally skew-symmetric torsion also have an important role.
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References
Bejan, C.L., Crasmareanu, M.: Conjugate connections with respect to a quadratic endomorphism and duality. Filomat 30(9), 2367–2374 (2016)
Bismut, J.M.: A local index theorem for non Kähler manifolds. Math. Ann. 284, 681–699 (1989)
Chern, S.S.: Characteristic classes of Hermitian manifolds. Ann. Math. 47, 85–121 (1946)
Cruceanu, V.: Almost product bicomplex structures on manifolds. An. St. Univ. A. I. Cuza Iasi 51(1), 99–118 (2005)
Cruceanu, V.: On almost biproduct complex manifolds. An. St. Univ. A. I. Cuza Iasi 52(1), 6–24 (2006)
Cruceanu, V., Etayo, F.: On almost para-Hermitian manifolds. Algebras Groups Geom. 16(1), 47–61 (1999)
Etayo, F., Santamaría, R.: Distinguished connections on \((J^{2}=\pm 1)\)-metric manifolds. Arch. Math. Brno 52(3), 159–203 (2016)
Etayo, F., Santamaría, R.: The well adapted connection of a \((J^2=\pm 1)\)-metric manifold. RACSAM 111(2), 355–375 (2017)
Fei, T., Yau, S.-T.: Invariant solutions to the Strominger system on complex Lie groups and their quotients. Commun. Math. Phys. 338, 1183–1195 (2015)
Fernández, M., Ivanov, S., Ugarte, L., Vassilev, D.: Non-Kaehler heterotic string solutions with non-zero fluxes and non-constant dilaton. J. High Energy Phys. 2014, 73 (2014)
Friedrich, T., Ivanov, S.: Parallel spinors and connections with skew-symmetric torsion in string theory. Asian J. Math. 6(2), 303–335 (2002)
Gauduchon, P.: Hermitian connections and Dirac operators. Boll. Un. Mat. Ital. B (7) 11(2, suppl.), 257–288 (1997)
Gray, A., Barros, M., Naveira, A.M., Vanhecke, L.: The Chern numbers of holomorphic vector bundles and formally holomorphic connections of complex vector bundles over almost-complex manifolds. J. Reine Angew. Math. 314, 84–98 (1980)
Gray, A., Hervella, L.M.: The sixteen classes of almost Hermitian manifolds and their linear invariants. Ann. Mat. Pura Appl. 123(1), 35–58 (1980)
Ivanov, S., Zamkovoy, S.: ParaHermitian and paraquaternionic manifolds. Differ. Geom. Appl. 23, 205–234 (2005)
Mekerov, D.: On the geometry of the connection with totally skew-symmetric torsion on almost complex manifolds with Norden metric. Comptes Rendus Acad. Bulg. Sci. 63(1), 19–28 (2010)
Mekerov, D., Manev, M.: Natural connection with totally skew-symmetric torsion on Riemann almost product manifolds. Int. J. Geom. Methods Mod. Phys. 9(1), 1250003 (2012)
Otal, A., Ugarte, L., Villacampa, R.: Invariant solutions to the Strominger system and the heterotic equations of motion. Nucl. Phys. B 920, 442–474 (2017)
Strominger, A.: Superstrings with torsion. Nucl. Phys. B 274, 253–284 (1986)
Teofilova, M.: Almost complex connections on almost complex manifolds with Norden metric. In: Sekigawa, K., Gerdjikov, V.S., Dimiev, S. (eds.) Trends in Differential Geometry, Complex Analysis and Mathematical Physics, pp. 231–240. World Scientific, Singapore (2009)
Acknowledgements
The authors are very grateful to Professors Mircea Crasmareanu, Luis Ugarte and Raquel Villacampa for their valuable and constructive comments that improved the manuscript. The authors are also very grateful to the referees for their careful reading and valuable suggestions.
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Etayo, F., deFrancisco, A. & Santamaría, R. The Chern Connection of a \((J^{2}=\pm 1)\)-Metric Manifold of Class \({\mathcal {G}}_{1}\). Mediterr. J. Math. 15, 157 (2018). https://doi.org/10.1007/s00009-018-1207-8
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DOI: https://doi.org/10.1007/s00009-018-1207-8