Some properties of almost hermitian metric pseudo f-manifolds and some applications on walker 4-manifolds

In this thesis, firstly, (ℳ2𝑛 , 𝑓, ℊ) almost Hermitian metric pseudo 𝑓 − manifolds is defined.On (ℳ2𝑛 , 𝑓, ℊ), definitions of first type special connection ∇̃, second type special connection ∇̅, 𝑓^2 −metric preserving connection ̂∇, 𝑓^2 −preserving connections ̿∇ and ∇̿^0 are given under some conditions. Torsion tensors 𝑇̃ , 𝑇̅ and 𝑇̂ of ∇̃, ∇̅ and ∇̂ connections are calculated, respectively. Finally, properties of almost Hermitian metric pseudo 𝑓 −structures are investigated on 4 −dimensional Walker manifolds (ℳ4 , ℊ) and the necessary and sufficient condition for the integrability of these structures is obtained. The necessary and sufficient condition is obtained for the triple (ℳ4 , 𝑓 , ℊ𝑊) to be a Hermitian metric pseudo 𝑓 −Kähler Walker manifold. The Riemannian curvature tensor 𝑅^𝑊, Ricci tensor 𝑅𝑖𝑐^𝑊 and scalar curvature 𝔰^𝑊 of the Walker 4 −manifold (ℳ4 , 𝑓 , ℊ𝑊) based on almost Hermitian metric pseudo 𝑓 −structure are calculated.