Geodesic
The structure of almost Hermitian structures of total space of principal fiber $T^1$-bundle with flat connection over some classes of almost contact metric manifolds
Čebyševskij sbornik, Tome 18 (2017) no. 2, pp. 183-194

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In paper we studied almost Hermitian structures of total space of principal fiber $T^1$-bundle 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 $W_2 \oplus W_4$ 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 $W_4$ 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 $G_1$ almost Hermitian structure, and over nearly cosymplectic manifolds almost Hermitian structure is $G_2$ almost Hermitian structure. Bibliography: 15 titles.
Keywords: principal fiber $T^1$-bundle, almost contact metric structure, almost Hermitian structure, Lee's form, local conformity. 
@article{CHEB_2017_18_2_a10,
     author = {I. A. Petrov},
     title = {The structure of almost {Hermitian} structures of total space of principal fiber $T^1$-bundle with flat connection over some classes of almost contact metric manifolds},
     journal = {\v{C}eby\v{s}evskij sbornik},
     pages = {183--194},
     publisher = {mathdoc},
     volume = {18},
     number = {2},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CHEB_2017_18_2_a10/}
}
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I. A. Petrov. The structure of almost Hermitian structures of total space of principal fiber $T^1$-bundle with flat connection over some classes of almost contact metric manifolds. Čebyševskij sbornik, Tome 18 (2017) no. 2, pp. 183-194. http://geodesic.mathdoc.fr/item/CHEB_2017_18_2_a10/