THE STRUCTURE OF ALMOST HERMITIAN STRUCTURES OF TOTAL SPACE OF PRINCIPAL FIBER T1-BUNDLE WITH FLAT CONNECTION OVER SOME CLASSES OF ALMOST CONTACT METRIC MANIFOLDS

Abstract

In paper we studied almost Hermitian structures of total space of principal fiber T¹bundle with flat connection over some classes of almost contact metric manifolds, such as contact, K−contact, Sasakian, normal, cosymplectic, nearly cosymplectic, exactly cosymplectic and weakly cosymplectic manifolds. Over contact and K−contact manifolds almost Hermitian structure belongs to the W₂⊕W₄ class. Lee’s form is different from the form of the flat connection by constant factor, equal to −2. Moreover, dual Lee’s vector field is different from some vector field from vertical distribution by the same constant factor. Also, this almost Hermitian structure is local conformal almost Kahlerian. Over Sasakian manifolds almost Hermitian structure belongs to the W₄ class. Lee’s form is different from the form of the flat connection by constant factor, equal to 2. Moreover, dual Lee’s vector field also is different from some vector field from vertical distribution by the same constant factor. Over weakly cosymplectic manifolds almost Hermitian structure is semiKahlerian. Lee’s form and dual Lee’s vector field are identically zero. Over cosymplectic manifolds almost Hermitian structure is Kahlerian. Also, Lee’s form and dual Lee’s vector field are identically zero. Over normal manifolds almost Hermitian structure is Hermitian. Over exactly cosymplectic manifolds almost Hermitian structure is G₁ almost Hermitian structure, and over nearly cosymplectic manifolds almost Hermitian structure is G₂ almost Hermitian structure.