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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.

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