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Mathematical Problems in Engineering
Volume 2009 (2009), Article ID 621625, 16 pages
Research Article

Warped Product Semi-Invariant Submanifolds in Almost Paracontact Riemannian Manifolds

Department of Mathematics, Faculty of Arts and Sciences, Gaziosmanpaşa University, 60250 Tokat, Turkey

Received 2 December 2008; Revised 5 May 2009; Accepted 14 July 2009

Academic Editor: Fernando Lobo Pereira

Copyright © 2009 Mehmet Atçeken. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We show that there exist no proper warped product semi-invariant submanifolds in almost paracontact Riemannian manifolds such that totally geodesic submanifold and totally umbilical submanifold of the warped product are invariant and anti-invariant, respectively. Therefore, we consider warped product semi-invariant submanifolds in the form 𝑁 = 𝑁 × 𝑓 𝑁 𝑇 by reversing two factor manifolds 𝑁 𝑇 and 𝑁 . We prove several fundamental properties of warped product semi-invariant submanifolds in an almost paracontact Riemannian manifold and establish a general inequality for an arbitrary warped product semi-invariant submanifold. After then, we investigate warped product semi-invariant submanifolds in a general almost paracontact Riemannian manifold which satisfy the equality case of the inequality.