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On conformal and quasi-conformal curvature tensors of an \(N(k)\)-quasi Einstein manifold. (English) Zbl 1252.53057

The authors study quasi-Einstein manifolds, i.e., Riemanian manifold whose Ricci tensor satisfies \(S=ag+b\eta\otimes\eta\), where \(\eta=g(\xi,.)\) and \(||\xi||=1\) and \(a,b\in C^{\infty}(M)\). Such a manifold is called an \(N(k)\)-quasi-Einstein manifold if \(\xi\in N(k)=\{Z\in TM: R(X,Y)Z=k(g(Y,Z)X-g(X,Z)Y)\}\) for some \(k\in C^{\infty}(M)\). The authors describes \(N(k)\)-quasi-Einstein manifolds satisfying the conditions \(C(\xi,X).S=0,\bar C(\xi,X).S=0,\;P(\xi,X).S=0,\;\bar P(\xi,X).S=0\), where \(C,\bar C, P,\bar P\) denote the conformal curvature tensor, the quasi-conformal curvature tensor, the projective curvature tensor and pseudo-projective curvature tensor, respectively.

MSC:

53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53B20 Local Riemannian geometry
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