Vol.3, No 14, 2003 pp. 821-842
UDC 514.82+514.74
Invited Paper

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ON QUASI EINSTEIN AND GENERALIZED
QUASI EINSTEIN MANIFOLDS
Sarbari Guha
Dept. of Physics, St. Xavier's College, 30 Park Street, Kolkata 700 016
E-mail: sarbariguha@rediffmail.com

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Abstract.  Recently, Prof. M. C. Chaki introduced the notion of a quasi Einstein manifold [1], denoted by (QE)n , whose Ricci tensor S of type (0,2) is not identically zero and satisfies the condition :
S(X,Y) = a g(X,Y) + b A(X)A(Y)
where a, b are scalars of which b?0 and A is a non-zero 1-form such that
g(X,U) = A(X)
for all vector fields X, U being a unit vector field.
If the existence of a 4-dimensional Lorentz manifold is established whose Ricci tensor is of the form given above, then it is found that such a space-time represents a perfect fluid space-time in cosmology.
Investigations by Karcher [2] and others have revealed that a conformally flat perfect fluid space-time has the geometric structure of quasi-constant curvature. It is found that a manifold of quasi-constant curvature is a natural sub-class of quasi Einstein manifold. Investigations on quasi Einstein manifolds help us to have a deeper understanding of the global character of the universe [3] including the topology. Consequently, we can study the nature of the singularities defined from a differential geometric standpoint.
 In a subsequent paper [4], Prof. Chaki introduced the generalized quasi Einstein manifolds denoted by G(QE)n. Chen and Yano [5] had introduced the notion of a manifold of quasi-constant curvature denoted by (QC)n. a generalization of a manifold of quasi-constant curvature, called a manifold of generalized quasi-constant curvature, denoted by G(QC)n, has been done by Prof. Chaki [4]. This is necessary for the study of G(QE)n. It is found that every G(QC)n (n 3) is a G(QE)n, while every G(QC)n (n?3) is a conformally flat G(QE)n. The importance of a G(QE)n lies in the fact that such a 4-dimensional semi-Riemannian manifold is relevant to the study of a general relativistic fluid space-time admitting heat flux [6]. The global properties of such a space-time is under investigation. Study of space-times admitting fluid viscosity and electromagnetic fields require further generalization of the Ricci tensor and is under process.

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KVAZI-EINSTEIN-OVE I GENERALISANE
KVAZI-EINSTEIN-OVE MNOGOSTRUKOSTI Profesor M.C. Chaki je uveo pojam kvazi-Eustein ova mnogostrukost [1], obeleživši je sa  , čiji je RICCI-jev tenzor S tipa (0,2) i nije identički jednak nulii zadovoljava sledeći uslov:
S(X,Y) = a g(X,Y) + b A(X)A(Y)
gde su  skalari od kojih je  and A je nenulta 1-forma takva da je
g(X,U) = A(X)
za sva vektorska polja X, U je jedinični vektor polja.
Ako postoji 4-dimenzionalna Lorencova mnogostrukost čiji je Ricci-jev tenzor u obliku prethodne forme, tada postoji takva prostor-vreme, koja predstavlja prostor-vreme idealnog fluida  u kosmologiji.
Istraživanja Karcher-a [2] i drugih su pokazala konformni ravni prostor-vreme idealnog fluida imaju geometrijsku strukturu kvazi-koonstantne krivine.  Utvrdjeno je da da mnogostrukost kvazi-konstantne strukture je prirodna podklasa kvazi-Eunstein-ove mnogostrukosti. Istraživanja kvazi-Eunstein-ovih mnogostrukosti nam pomažu da dublje razumemo globalni karakter univerzuma [3] uključujući i topologiju. Kao posledica toga, u ovom radu je studirana priroda singulariteta definitnih formi sa stanovišta diferencijalne geometrije. Istraživanja mnogostrukosti prostor-vreme omogućavaju uvodjenje viskoznog fluida i eletromagnetskog  polja, kao i buduće generalizacije Ricci-jevih tenzora, koji su u toku.