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.

Voir la notice de l'article provenant de la source Math-Net.Ru

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/}
}
TY  - JOUR
AU  - I. A. Petrov
TI  - 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
JO  - Čebyševskij sbornik
PY  - 2017
SP  - 183
EP  - 194
VL  - 18
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/CHEB_2017_18_2_a10/
LA  - ru
ID  - CHEB_2017_18_2_a10
ER  - 
%0 Journal Article
%A I. A. Petrov
%T 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
%J Čebyševskij sbornik
%D 2017
%P 183-194
%V 18
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/CHEB_2017_18_2_a10/
%G ru
%F CHEB_2017_18_2_a10
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/

[1] Galaev S. V., Shevchova Y. V., “Almost contact metric structures”, Izv. Sarat. un-ta, 15:2 (2015), 136–141 | Zbl

[2] Dondukova N. N., “Projective invariant of cosymplectic manifolds”, Vest. Bur. gos. un-ta, 2011, no. 9, 171–175

[3] Banaru M. B., “On the typical number of weakly cosymplectic hypersurfaces of approximately Kahler manifolds”, Fundament. i prikl. matem., 8:2 (2002), 357–364

[4] Vinberg E. B., Onishchik A. L., Seminar on Lee's groups and algebraic groups, Nauka, M., 1988

[5] Kirichenko V. F., Differential geometric structure on manifolds, Odessa, 2013

[6] Savinov A. V., “Canonical toroidal fibre bundle under an odd-dimensional base”, Vestnik SamGU, 2003, no. 2(28), 57–79

[7] Gray A., Hervella L. M., “The Sixteen Classes of Almost Hermitian Manifolds and Their Linear Invariants”, Annali di Matematica pura ed applicata, CXXIII:IV (1980), 35–38 | DOI | MR

[8] Kirichenko V. F., “Differential geometry of principal toroidal fiber bundles”, Fundamentalnaya i prikladnaya matematika, 6:4 (2000), 1095–1120 | Zbl

[9] Blair D., “Contact manifolds in geometry”, Lecture Notes in Math., 509, 1976, 1–145 | DOI | MR

[10] Kobayashi S., “Principal fibre bundle with the 1-dimensional toroidal group”, Tohoku Math. J., 8 (1956), 29–45 | DOI | MR | Zbl

[11] Watanabe Y., “Riemannian metrics on principal circle bundles over lokally symmetric Kahlerian manifolds”, Kodai Math. J., 5 (1982), 111–121 | DOI | MR | Zbl

[12] Borisovskii I. P., “On the geometry of principal T1-bundles over Hodge manifolds”, Math. Notes, 64:6 (1998), 714–718 | DOI | DOI | MR | Zbl

[13] Ignatochkina L. A., “Generalization for transformations of T1-bundle which induced by conformal transformations of their base”, Sb. Math., 202:5 (2011), 665–682 | DOI | DOI | MR | Zbl

[14] Ignatochkina L. A., Analysis on the manifolds, MPSU, M., 2016

[15] Ignatochkina L A., A short guide to the action on principal fiber bundles and the method of the associated $G$-structure, M., 2014