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ISRN Geometry
Volume 2012 (2012), Article ID 421384, 13 pages
http://dx.doi.org/10.5402/2012/421384
Research Article

Ricci Solitons in 𝛼-Sasakian Manifolds

Department of Mathematics, Kuvempu University, Shankaraghatta 577 451, India

Received 20 April 2012; Accepted 28 May 2012

Academic Editors: F. P. Schuller and I. Strachan

Copyright © 2012 Gurupadavva Ingalahalli and C. S. Bagewadi. 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.

Abstract

We study Ricci solitons in 𝛼-Sasakian manifolds. It is shown that a symmetric parallel second order-covariant tensor in a 𝛼-Sasakian manifold is a constant multiple of the metric tensor. Using this, it is shown that if 𝑉𝑔+2𝑆 is parallel where 𝑉 is a given vector field, then (𝑔,𝑉,𝜆) is Ricci soliton. Further, by virtue of this result, Ricci solitons for 𝑛-dimensional 𝛼-Sasakian manifolds are obtained. Next, Ricci solitons for 3-dimensional 𝛼-Sasakian manifolds are discussed with an example.