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\(\xi\)-null geodesic gradient vector fields on a Lorentzian para-Sasakian manifold. (English) Zbl 0828.53042

Lorentzian para-Sasakian manifolds \(M(\varphi, \xi, \eta, g)\) were defined by K. Matsumoto and I. Mihai in [Tensor, New Ser. 47, No. 2, 189-197 (1988; Zbl 0679.53034)]. Generally speaking, they are \((2m +1)\)-dimensional differentiable Lorentzian manifolds endowed with a kind of almost paracontact structures. In the present paper, the authors assume that such a manifold \(M\) admits a special vector field, say \(U\), satisfying the analytical condition \(\nabla_X U = \lambda X + \eta(X) U + g(X, U) \xi\) for any vector field \(X\) on \(M\), \(\lambda\) being a scalar function on \(M\). It is proved that the manifold is locally a product \(M_U \times M^\perp_U\), where \(M_U\) is a totally geodesic surface of scalar curvature \((-1)\) which is tangent to \(U\) and \(\xi\), and \(M^\perp_U\) is a totally umbilical submanifold of \(M\) of codimension 2. Certain additional curvature consequences for \(M\) are also derived.

MSC:

53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics

Citations:

Zbl 0679.53034
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