Abstract
In the present paper firstly, we introduce classes of anti-paraK\"{a}hler-Codazzi manifolds and we discuss the problem of integrability for almost paracomplex structures on thes manifolds. Secondly, we introduce a new classes of anti-paraHermitian manifolds associated with these anti-paraHermitian metric connections with torsion, we look for the conditions in which it becomes are anti-paraK\"{a}hler manifolds or anti-paraK\"{a}hler-Codazzi manifolds.
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