Matsumoto, Koji; Mihai, Ion; Rosca, Radu \(\xi\)-null geodesic gradient vector fields on a Lorentzian para-Sasakian manifold. (English) Zbl 0828.53042 J. Korean Math. Soc. 32, No. 1, 17-31 (1995). 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. Reviewer: Z.Olszak (Wrocław) Cited in 13 Documents MSC: 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.) 53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics Keywords:Lorentzian manifold; para-Sasakian manifolds Citations:Zbl 0679.53034 PDFBibTeX XMLCite \textit{K. Matsumoto} et al., J. Korean Math. Soc. 32, No. 1, 17--31 (1995; Zbl 0828.53042)