Vanishing quasi-conformal curvature tensor of a certain class of almost contact metric manifolds
Main Article Content
Abstract
This paper focuses on the essential role of the quasi-conformal curvature
tensor in developing the theoretical foundation and applications of quasi-contact
metric geometry. The quasi-conformal curvature tensor of NC10-manifolds using the
space associated with G structures (AGS-space) has been investigated. By deriving
explicit expressions for the tensor components, the necessary and sufficient conditions
for considering the quasi-conformal NC10-manifold as flat quasi-conformal have been
determined. Noteworthy, it indicated that any flat quasi-conformal NC10-manifold
is locally isometric to the product of the Kähler manifold and the real line, thus
connecting abstract tensor properties to classical geometric structures. Moreover, the
conditions for the NC10-manifold that yield the η-Einstein structure are established.
New analogues of Gray’s identities for the quasi-conformal curvature tensor have
been introduced for NC10-manifold.
Article Details
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
References
[1] H. Abood and M. Abass, A Study of New Class of Almost Contact Metric Manifolds of Kenmotsu Type, Tamkang J. Math., 52(2), 253-266, 2021.
[2] Abood H. M., Al-Hussaini F. H., On the Conharmonic Curvature Tensor of A Locally Conformal Almost Cosymplectic Manifold, Commun. Korean Math. Soc., 35(1), 269-278, 2020.
[3] Al-Hussaini F. H. and Abood H. M., Quaisi Invariant Conharmonic Tensor of Special Classes of A Locally Conformal Almost Cosymplectic Manifold, Vestnic Udmurt skogo Universiteta. Matematika. Mekhanika Komp’uternye Nauki, 30(2), 147-157, 2020.
[4] Al-Hussaini F. H., Rustanov A. R., Abood H. M., Vanishing Conharmonic Tensor of Normal Locally Conformal Almost Cosymplectic Manifold, Commentationes Mathematicae Universitatis Caroline, 1, 93-104, 2020.
[5] AlHusseini F. H., Abood H. M., Quasi Sasakian Manifold Endowed with Vanishing Pseudo Quasi Conformal Curvature Tensor, MethodsX, 15, 2025.
[6] Blair D.E, The theory of quasi-Sasakian structures, J. Differential Geometry, 1, 331-345, 1967.
[7] Blair D. E. , Riemannian Geometry of Contact and Symplectic Manifolds, in Progr.Math. Birkhauser, Boston, P. 203, 2002.
[8] Cartan É., Riemannian Geometry in an Orthogonal Frame, From lectures Delivered by É. Lie Cartan at the Sorbonne 1926-27, Izdat. Moskov. Univ., Moscow, 1960; World Sci., Singapore, 2001.
[9] Chinea D. and Gonzalez C., A classification of almost contact metric manifolds, Annali di Matematica Pura ed Applicata, 156(1), 15-36, 1990.
[10] Goldberg S.I. and Yano K., Integrabilty of almost cosymplectic structures, Pacific Journal of Mathematics, 31, 373-382, 1969.
[11] Kirichenko V.F., The method of generalization of Hermitian geometry in the almost Hermitian contact manifold, Journal of Soviet Mathematics, 42, 1885-1919, 1998.
[12] Kirichenko V.F., Differential - Geometry structures on manifolds, Second edition, expanded. Odessa, Printing House, P. 458, 2013.
[13] Kirichenko V. F. and Rustanov A. R., Differential Geometry of quasi Sasakian manifolds, Sbornik: Mathematics, 193(8), 1173-1201, 2002.
[14] Kirichenko V.F. and Kharitonova S. V., On the geometry of normal locally conformal almost cosymplectic manifolds, Mathematical Notes, 91 (1), 40-53, 2012.
[15] Rustanov, A. R., Integrability Properties of NC10 -Manifolds, MATEMATEKA & MEXANEKA,5(20), 32-38, 2017.
[16] Rustanov, A. R., Kazakova O. N., Kharitonova S. V., Contact analogs of Gray’s identoties of NC10 -Manifold, Siberian Electronic Mathematical Reports, 5, 823-828, 2018.
[17] Yano K. and Sewaki S., Riemannian Manifolds Admitting a Conformal Transformation Group, J. Diff. Geom. 2, 161-184, 1968.