Izvorni znanstveni članak

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

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

David N Pham orcid.org/0000-0001-5615-0719 ; Department of Mathematics & Computer Science, Queensborough C. College, City University of New York, Bayside, NY 11364, USA
Fei Ye ; Department of Mathematics & Computer Science, Queensborough C. College, City University of New York, Bayside, NY 11364, USA

Preuzmi JATS datoteku

Sažetak

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 $\mathfrak{g}:=\text{Lie}(G)$. 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 $\mathfrak{h}:=\text{Lie}(H)$. 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 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.

Ključne riječi

Almost Hermitian manifolds, Hermitian connections, Gauduchon connections, Lie groups

Hrčak ID:

325177

URI

https://hrcak.srce.hr/325177

Datum izdavanja:

3.4.2025.

Posjeta: 145 *