Left-invariant Hermitian connections on Lie groups with almost Hermitian structures

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.