International Scholarly Research Network ISRN Mathematical Physics Volume 2012, Article ID 251460, 9 pages doi:10.5402/2012/251460 Research Article Bianchi Type—IX Barotropic Fluid Model with Time-Dependent Displacement Vector in Lyra Geometry Raj Bali, Rajendra Vadhwani, and Jagdish Prasad Dhanka Department of Mathematics, University of Rajasthan, Jaipur 302004, India Correspondence should be addressed to Raj Bali, balir5@yahoo.co.in Received 17 September 2011; Accepted 19 October 2011 Academic Editors: J. Biˇ c´ ak and D. Singleton Copyright q 2012 Raj Bali et al. 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. Bianchi Type IX barotropic fluid cosmological model in the frame work of Lyra geometry is investigated. To get the deterministic model of the universe, it is assumed that shear σ is proportional to expansion θ. This leads to a  b n , where a and b are metric potentials and n is a constant. To get the results in terms of cosmic time t, we have also considered a special case γ  0 dust filled universe and n  2. We find that the model starts with a big bang initially and the displacement vector β is initially large but decreases due to lapse of time. The models ds 2  −T 6 /3N 2 /20 − 1 − γ /45γ  7T 8 − γ/5γ  4T 6 dT 2  T 4 dX 2  T 2 dY 2 T 2 sin 2 Y  T 4 cos 2 Y dZ 2 − 2T 4 cos Y dX dZ and ds 2  −dτ 2    21/5N sin2/ √ 7τ dx 2   21/5N sin2/ √ 7τ  1/2 dy 2   21/5N sin2/ √ 7τ  1/2 sin 2 y    21/5N sin2/ √ 7τ cos 2 ydz 2 − 2  21/5N sin2/ √ 7τ cos ydx dz have point-type singu- larity at T  0 and τ  0, respectively. The physical and geometrical aspects of the models are also discussed. 1. Introduction Bianchi Type IX space-time is the generalization of FRW model with positive curvature. Bianchi Types cosmological models create more interest in the study, because familiar models like Robertson-Walker model 1, the de-Sitter universe 2, Taub-NUT 3, 4 space times are of Bianchi Type IX space-time. The solutions 3, 4 allow expansion, rotation, and shear. Vaidya and Patel 5 have obtained the solution for spatially homogeneous Bianchi Type IX space time and have given a general scheme for the derivation of exact solutions of Einstein’s field equations corresponding to a perfect fluid and pure radiation field. Bianchi Type IX space times are also studied by many research workers namely Krori et al. 6, Chakraborty and Nandy 7, Chakraborty 8, and Bali and Upadhaya 9.