%0 Journal Article
%T Paracomplex Paracontact Pseudo-Riemannian Submersions
%A S. S. Shukla
%A Uma Shankar Verma
%J Geometry
%D 2014
%I Hindawi Publishing Corporation
%R 10.1155/2014/616487
%X We introduce the notion of paracomplex paracontact pseudo-Riemannian submersions from almost para-Hermitian manifolds onto almost paracontact metric manifolds. We discuss the transference of structures on total manifolds and base manifolds and provide some examples. We also obtain the integrability condition of horizontal distribution and investigate curvature properties under such submersions. 1. Introduction The theory of Riemannian submersion was introduced by O¡¯Neill [1, 2] and Gray [3]. It is known that the applications of such Riemannian submersion are extensively used in Kaluza-Klein theories [4, 5], Yang-Mill equations [6, 7], the theory of robotics [8], and supergravity and superstring theories [5, 9]. There is detailed literature on the Riemannian submersion with suitable smooth surjective map followed by different conditions applied to total space and on the fibres of surjective map. The Riemannian submersions between almost Hermitian manifolds have been studied by Watson [10]. The Riemannian submersions between almost contact manifolds were studied by Chinea [11]. He also concluded that if is an almost Hermitian manifold with structure ( ) and is an almost contact metric manifold with structure ( ), then there does not exist a Riemannian submersion which commutes with the structures on and ; that is, we cannot have the condition . Chinea also defined the Riemannian submersion between almost complex manifolds and almost contact manifolds and studied some properties and interrelations between them [12]. In [13], G¨¹nd¨¹zalp and Sahin gave the concept of paracontact paracomplex semi-Riemannian submersion between almost paracontact metric manifolds and almost para-Hermitian manifolds submersion giving an example and studied some geometric properties of such submersions. An almost paracontact structure on a differentiable manifold was introduced by Sato [14], which is an analogue of an almost contact structure and is closely related to almost product structure. An almost contact manifold is always odd dimensional but an almost paracontact manifold could be even dimensional as well. The paracomplex geometry has been studied since the first papers by Rashevskij [15], Libermann [16], and Patterson [17] until now, from several different points of view. The subject has applications to several topics such as negatively curved manifolds, mechanics, elliptic geometry, and pseudo-Riemannian space forms. Paracomplex and paracontact geometries are topics with many analogies and also with differences with complex and contact geometries. This motivated us to
%U http://www.hindawi.com/journals/geometry/2014/616487/