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On \(\eta\)-Einstein Sasakian manifolds. (English) Zbl 0667.53040

Let M(\(\phi\),\(\eta\),\(\xi\),g) be an n-dimensional Sasakian manifold with structure tensor fields \(\phi\), \(\eta\), \(\xi\) \(\phi^ 2=-Id+\eta \otimes \xi)\) and let \({\mathcal S}\) be the Ricci tensor of the covariant differential operator \(\nabla\). M is defined as an \(\eta\)-Einsteinian Sasakian manifold [K. Yano and M. Kon, CR submanifolds of Kaehlerian and Sasakian manifolds (1983; Zbl 0496.53037)] iff \(S(X,Y)=ag(X,Y)+b\eta (X)\eta (Y);\) a, b are constants, X, Y vector fields. In the present paper the author proves the following theorems: (i) An \(\eta\)-Einstein Sasakian manifold is of constant scalar curvature if and only if \[ (\nabla_ XS)(Y,Z)+(\nabla_ YS)(Z,X)+(\nabla_ ZS)(X,Y)=0. \] (ii) If in an \(\eta\)-Einstein manifold the relation \(R(X,Y)S=0\) (R denoting the curvature tensor field) holds, then the manifold is an Einstein manifold (i.e. \(b=0)\). (iii) If an \(\eta\)- Einstein manifold of constant scalar curvature the Ricci tensor is a Codazzi tensor then the manifold is Ricci symmetric.
Reviewer: R.Rosca

MSC:

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

Citations:

Zbl 0496.53037
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