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 ( (<middle dot>, <middle dot>), J). The space of left-invariant Hermitian connections is shown to be in one-to-one correspondence with the space n (1,1) g & lowast; 0 g of left-invariant 2-forms of type (1,1) (with respect to J) with values in 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 x 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 h := Lie(H). When H is compact, it is shown that H x A admits a left-invariant (strictly) almost Hermitian structure ( (<middle dot>, <middle dot>), 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 ( (<middle dot>, <middle dot>), J) on H x A is shown to satisfy the strong Ka<spacing diaeresis>hler with torsion condition. Furthermore, the affine line of Gauduchon connections on H x A with the aforementioned almost Hermitian structure is also shown to contain a (nontrivial) flat connection.