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In this paper, we first define the concept of paracontact semi-Riemannian submersions between almost paracontact metric manifolds, then we provide an example and show that the vertical and horizontal distributions of such submersions are invariant with respect to the almost paracontact structure of the total manifold. | Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ Research Article Turk J Math (2013) 37: 114 – 128 ¨ ITAK ˙ c TUB doi:10.3906/mat-1103-10 Paracontact semi-Riemannian submersions 1 ¨ ¨ ˙ 2,∗ Yılmaz GUND UZALP , Bayram S ¸ AHIN Faculty of Education, Dicle University, Diyarbakır, Turkey 2 ˙ on¨ Department of Mathematics, In¨ u University, 44280, Malatya, Turkey 1 Received: 04.03.2011 • Accepted: 08.10.2011 • Published Online: 17.12.2012 • Printed: 14.01.2013 Abstract: In this paper, we first define the concept of paracontact semi-Riemannian submersions between almost paracontact metric manifolds, then we provide an example and show that the vertical and horizontal distributions of such submersions are invariant with respect to the almost paracontact structure of the total manifold. The study is focused on fundamental properties and the transference of structures defined on the total manifold. Moreover, we obtain various properties of the O’Neill’s tensors for such submersions and find the integrability of the horizontal distribution. We also find necessary and sufficient conditions for a paracontact semi-Riemannian submersion to be totally geodesic. Finally, we obtain curvature relations between the base manifold and the total manifold. Key words: Almost paracontact metric manifold, semi-Riemannian submersion, paracontact semi-Riemannian submersion 1. Introduction The theory of Riemannian submersion was introduced by O’Neill and Gray in [11] and [6], respectively. Presently, there is an extensive literature on the Riemannian submersions with different conditions imposed on the total space and on the fibres. Semi-Riemannian submersions were introduced by O’Neill in his book [12]. Later, Riemannian submersions were considered between almost complex manifolds by Watson in [13] under the name of almost Hermitian submersion. He showed that if the total manifold is a K¨ ahler manifold, the base manifold is also a K¨ahler manifold. Riemannian .
