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On almost cosymplectic manifolds with the structure vector field \(\xi\) belonging to the \(\kappa\)-nullity distribution. (English) Zbl 0981.53075
Summary: Let \(M\) be an almost cosymplectic manifold whose structure vector field \(\xi\) belongs to the so-called \(k\)-nullity distribution. Then \(k\leq 0\) and the local structure of \(M\) is completely described. For a positive constant \(\lambda\), it can be defined a Lie group \(G_\lambda\) endowed with a left invariant almost cosymplectic structure whose structure vector field \(\xi\) belongs to the \((-\lambda^2)\)-nullity distribution. It is proved that if \(k<0\), then \(M\) is locally isomorphic to \(G_\lambda\) with \(k=-\lambda^2\); especially \(M\) is a locally homogeneous Riemannian manifold. We also show that in this case the manifold \(M\) is Ricci-pseudosymmetric. \(k=0\) if and only if \(M\) is locally a product of an open interval and an almost Kähler manifold.

53D15 Almost contact and almost symplectic manifolds
53C20 Global Riemannian geometry, including pinching
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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