Glasnik Matematicki, Vol. 59, No. 2 (2024), 417-460. \( \)

LEFT-INVARIANT HERMITIAN CONNECTIONS ON LIE GROUPS WITH ALMOST HERMITIAN STRUCTURES

David N. Pham and Fei Ye

Department of Mathematics \(\&\) Computer Science, Queensborough C. College, City University of New York, Bayside, NY 11364, USA
e-mail:dpham90@gmail.com

Department of Mathematics \(\&\) Computer Science, Queensborough C. College, City University of New York, Bayside, NY 11364, USA
e-mail:feye@qcc.cuny.edu

Abstract.   Left-invariant Hermitian and Gauduchon connections are studied on an arbitrary Lie group \(G\) equipped with an arbitrary left-invariant almost Hermitian structure \((\langle\cdot,\cdot\rangle,J)\). The space of left-invariant Hermitian connections is shown to be in one-to-one correspondence with the space \(\wedge^{(1,1)}\mathfrak{g}^\ast\otimes \mathfrak{g}\) of left-invariant 2-forms of type (1,1) (with respect to \(J\)) with values in \(\mathfrak{g}:=\mbox{Lie}(G)\). Explicit formulas are obtained for the torsion components of every Hermitian and Gauduchon connection with respect to a convenient choice of left-invariant frame on \(G\). The curvature of Gauduchon connections is studied for the special case \(G=H\times A\), where \(H\) is an arbitrary \(n\)-dimensional Lie group, \(A\) is an arbitrary \(n\)-dimensional abelian Lie group, and the almost complex structure is totally real with respect to \(\mathfrak{h}:=\mbox{Lie}(H)\). When \(H\) is compact, it is shown that \(H\times A\) admits a left-invariant (strictly) almost Hermitian structure \((\langle\cdot,\cdot\rangle,J)\) such that the Gauduchon connection corresponding to the Strominger (or Bismut) connection in the integrable case is precisely the trivial left-invariant connection and, in addition, has totally skew-symmetric torsion. The almost Hermitian structure \((\langle\cdot,\cdot\rangle,J)\) on \(H\times A\) is shown to satisfy the strong Kähler with torsion condition. Furthermore, the affine line of Gauduchon connections on \(H\times A\) with the aforementioned almost Hermitian structure is also shown to contain a (nontrivial) flat connection.

2020 Mathematics Subject Classification.   53B05, 53C15, 32Q60

Key words and phrases.   Almost Hermitian manifolds, Hermitian connections, Gauduchon connections, Lie groups

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https://doi.org/10.3336/gm.59.2.08

References:

  1. M. M. Alexandrino and R. G. Bettiol, Introduction to Lie groups, isometric and adjoint actions and some generalizations, arXiv:0901.2374.

  2. A. Andrada, M. L. Barberis, I. G. Dotti and G. P. Ovando, Product structures on four dimensional solvable Lie algebras, Homology Homotopy Appl. 7 (2005), 9–37.
    MathSciNet

  3. A. Andrada and R. Villacampa, Bismut connection on Vaisman manifolds, Math. Z. 302 (2022), 1091–1126.
    MathSciNet    CrossRef

  4. J. M. Bismut, A local index theorem for non-Kähler manifolds, Math. Ann. 284 (1989), 681–699.
    MathSciNet    CrossRef

  5. S. S. Chern, Complex manifolds without potential theory, universitext, Springer-Verlag, New York-Heidelberg, 1979.
    MathSciNet    CrossRef

  6. R. Cleyton, J. Lauret and Y. S. Poon, Weak mirror symmetry of Lie algebras, J. Symplectic Geom. 8 (2010), 37–55.
    MathSciNet    CrossRef

  7. R. Cleyton, Y. S. Poon and G. P. Ovando, Weak mirror symmetry of complex symplectic Lie algebras, J. Geom. Phys. 61 (2011), 1553–1563.
    MathSciNet    CrossRef

  8. R. Campoamor-Stursberg and G. P. Ovando, Invariant complex structures on tangent and cotangent Lie groups of dimensions six, Osaka J. Math. 49 (2012), 489–513.
    MathSciNet

  9. A. Fino and A. Tomassini, A survey on strong KT structures, Bull. Math. Soc. Sci. Math. Roumanie, 52 (2009), 99–116.
    MathSciNet

  10. T. Friedrich and S. Ivanov, Parallel spinors and connections with skew-symmetric torsion in string theory, Asian J. Math. 6 (2002), 303–335.
    MathSciNet    CrossRef

  11. S. J. Gates Jr., C. M. Hull and M. Roček, Twisted multiplets and new supersymmetric nonlinear \( \sigma\)-models, Nuclear Phys. B, 248 (1984), 157–186.
    MathSciNet    CrossRef

  12. P. Gauduchon, Hermitian connections and Dirac operators, Boll. Un. Mat. Ital. B 7 (1997), 257–288.
    MathSciNet

  13. B. C. Hall, Lie groups, Lie algebras, and representations. An elementary introduction, Springer New York, 2004.
    MathSciNet    CrossRef

  14. R. A. Lafuente and J. Stanfield, Hermitian manifolds with flat Gauduchon connections, to appear in Ann. Scuola Norm-Sci.
    CrossRef

  15. A. Strominger, Superstrings with Torsion, Nuclear Phys. B 274 (1986), 253–284.
    MathSciNet    CrossRef

  16. L. Vezzoni, B. Yang and F. Zheng, Lie groups with flat Gauduchon connections, Math. Z. 293 (2019), 597–608.
    MathSciNet    CrossRef

  17. F. W. Warner, Foundations of differentiable manifolds and Lie groups, Springer-Verlag, New York-Berlin, 1983.
    MathSciNet    CrossRef

  18. B. Yang and F. Y. Zheng, On compact Hermitian manifolds with flat Gauduchon connections, Acta Math. Sin. 34 (2018), 1259–1268.
    MathSciNet    CrossRef
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