## 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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