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 $(⟨\cdot ,\cdot ⟩,J)$. 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 $\mathfrak{g}:=Lie(G)$. 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 $\mathfrak{h}:=Lie(H)$. When $H$ is compact, it is shown that $H\times A$ admits a left-invariant (strictly) almost Hermitian structure $(⟨\cdot ,\cdot ⟩,J)$ 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 $(⟨\cdot ,\cdot ⟩,J)$ on $H\times A$ is shown to satisfy the $\mathit{\text{strong}}\mathit{\text{K}}\ddot{\mathit{\text{a}}}\mathit{\text{hler}}\mathit{\text{with}}\mathit{\text{torsion}}$ 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.