A Unified Framework for Hermitian-Like and Anti-Hermitian-Like Geometries

This paper explores how certain generalized structures in differential geometry can be used to construct Kähler and anti-Kähler manifolds. In Hermitian-like settings, a specific type of tensor is introduced, and its antisymmetric component leads to the formation of Kähler-like geometries. In anti-Hermitian-like settings, the symmetric component of the similar tensor gives rise to anti-Kähler structures. The study consists of two main parts. The first part investigates almost Hermitian-like manifolds, where certain conditions such as the parallelism of a structural endomorphism and the vanishing of associated tensors help to characterize extended forms of Kähler geometry, including Hsu–Hermitian and Hsu–Kähler structures. The second part focuses on almost anti-Hermitian-like manifolds and demonstrates that, under appropriate compatibility with torsion-free connections, the geometry supports anti-Kähler and Hsu-B structures. These results together present a unified and systematic approach to generalizing classical complex geometric frameworks using differential and algebraic methods.