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Zbl 0828.53042
Matsumoto, Koji; Mihai, Ion; Rosca, Radu
$\xi$-null geodesic gradient vector fields on a Lorentzian para-Sasakian manifold.
(English)
[J] J. Korean Math. Soc. 32, No.1, 17-31 (1995). ISSN 0304-9914

Lorentzian para-Sasakian manifolds $M(\varphi, \xi, \eta, g)$ were defined by {\it K. Matsumoto} and {\it 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.
[Z.Olszak (Wrocław)]
MSC 2000:
*53C25 Special Riemannian manifolds
53C50 Lorentz manifolds, manifolds with indefinite metrics

Keywords: Lorentzian manifold; para-Sasakian manifolds

Citations: Zbl 0679.53034

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