Fortuné MASSAMBA, Tshikunguila TSHIKUNA-MATAMBA

Horizontally submersions of contact CR-submanifolds

In this paper, we discuss some geometric properties of almost contact metric submersions involving symplectic manifolds. We show that the structures of quasi-K-cosymplectic and quasi-Kenmotsu manifolds are related to (1, 2)-symplectic structures. For horizontally submersions of contact CR-submanifolds of quasi-K-cosymplectic and quasi-Kenmotsu manifolds, we study the principal characteristics and prove that their total spaces are CR-product. Curvature properties between curvatures of quasi-K-cosymplectic and quasi-Kenmotsu manifolds and the base spaces of such submersions are also established. We finally prove that, under a certain condition, the contact CR-submanifold of a quasi Kenmotsu manifold is locally a product of a totally geodesic leaf of an integrable horizontal distribution and a curve tangent to the normal distribution.

Anahtar Kelimeler:

CR-submanifold, almost Hermitian manifold, almost contact metric submersion, symplectic manifold, horizontal submersion

Horizontally submersions of contact CR-submanifolds

Keywords:

CR-submanifold, almost Hermitian manifold, almost contact metric submersion, symplectic manifold, horizontal submersion,

PDF

___

Bejancu A. CR -submanifolds of a Kaehler manifold. I. Proc Amer Math Soc 1978; 69: 135–142.
Bejancu A, Papaghiuc N. Semi-invariant submanifolds of a Sasakian manifold. An S ¸tint¸ Univ Al I Cuza Ia¸si Mat (NS) 1981; 27: 163–170.
Bejancu A., Papaghiuc N. Semi-invariant submanifolds of a Sasakian space form. Colloq Math 1984; 48: 77–88.
Blair DE. Riemannian Geometry of Contact and Symplectic Manifolds. Progress in Mathematics. New York, NY, USA: Birkh¨ auser, 2002.
Burel JM. Almost contact structures and harmonic maps with minimal fibres. Houston J Math 2004; 30: 393–411. Chen BY. CR -submanifolds of a Kaehler manifold I, II. J Differ Geom 1981; 16: 305–322, 493–509.
Chinea D. Quasi- K -cosymplectic submersions. Rend Circ Mat Palermo 1984; 33: 319–330.
Chinea D. Almost contact metric submersions. Rend Circ Mat Palermo 1985; 34: 89–104.
Chinea D. Harmonicity on maps between almost contact metric manifolds. Acta Math Hungar 2010; 126: 352–365. Chinea D, Gonzalez C. A classification of almost contact metric manifolds. Ann Mat Pura Appl 1990; 156: 15–36. Deshmukh S, Ghazal T, Hashem H. Submersions of CR -submanifolds on an almost Hermitian manifold. Yokohama Math J 1992; 40: 45–57.
Dragomir S, Ornea L. Locally Conformal Kaehler Geometry. Progress in Mathematics. Boston, MA, USA: Birkh¨ auser, 1998.
Gray A, Hervella LM. The sixteen classes of almost Hermitian manifolds and their linear invariants. Ann Mat Pura Appl 1980; 123: 35–58.
O’Neill O. The fundamental equations of a submersion. Michigan Math J 1966; 13: 459–469.
Papaghiuc N. Submersions of semi-invariant submanifolds of a Sasakian manifold. An Stint Univ Al I Cuza Iasi Sect I-a Mat 1989; 35: 281–288.
Sahin B. Horizontally conformal submersions of CR -submanifolds. Kodai Math J 2008; 31: 46–53.
Watson B. The differential geometry of two types of almost contact metric submersions. In: Rassias GM, editor. The. Mathematical Heritage of C. F. Gauss. Singapore: World Scientific Publishing, 1991, pp. 827–861.
Yano K., Kon M. Structures on Manifolds. Series in Pure Mathematics. Singapore: World Scientific Publishing, 19