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On an Einstein \(M\)-projective \(P\)-Sasakian manifolds. (English) Zbl 1201.53036

Summary: We prove that if in an Einstein \(P\)-Sasakian manifold \(R(T,X)\cdot W^*=0\) holds, then it is locally isometric with \(S^n(1)\). It is also proved that an \(n\)-dimensional \(\eta\) Einstein \(P\)-Sasakian manifold \(M\) satisfies \(W^*(T,X)\cdot R=0\) if an only if either \(M^n\), is locally isometric to the hyperbolic space \(H^n(-1)\) or the scalar curvature \(\mathfrak r\) of \(M^n\) is \(r=-n(n-1)\).

MSC:

53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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