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Lightlike hypersurfaces of indefinite Sasakian manifolds with parallel symmetric bilinear forms. (English) Zbl 1162.53017

Let \(\widetilde M\) be a manifold with a contact metric structure \((\varphi,\xi,\eta,\widetilde g)\). A normal contact metric manifold \(\widetilde M\) is called indefinite Sasakian if \(\nabla_ X\xi=\varphi X\) and \((\nabla_ X\varphi)Y=-\widetilde g(X,Y)\xi+\eta(Y)X\). Let \((\widetilde M,\widetilde g)\) be an \((n+2)\)-dimensional semi-Riemannian manifold of constant index \(0<\nu<n+2\), and let \((M,g)\) be its hypersurface. In the theory of nondegenerate hypersurfaces, the normal bundle has trivial intersection with the tangent bundle. If \(M\) is a light-like hypersurface, then the normal bundle \(TM^\perp\) is a rank-one distribution over \(M\) and coincides with the so-called radical distribution \(\text{Rad}(TM)=TM\cap TM^\perp\). The induced metric \(g\) on \(M\) is degenerate and has constant rank \(n\). A complementary bundle of \(\text{Rad}(TM)\) in \(TM\) is a rank \(n\) nondegenerate screen distribution \(S(TM)\) on \(M\). If the second fundamental form \(h\) of a hypersurface \(M\) of an indefinite Sasakian manifold \(\widetilde M(\varphi,\xi,\eta,\widetilde g)\) tangent to \(\xi\) vanishes, then \(M\) is called totally geodesic.
In this paper, the author studies properties of light-like hypersurfaces of indefinite Sasakian manifold which are tangent to the structure vector field \(\xi\). Some necessary and sufficient conditions for three classes of totally geodesic light-like hypersurfaces \(M\) of indefinite Sasakian space forms with parallel symmetric bilinear forms are presented. Also, it is shown that the geometry of light-like hypersurface \(M\) of indefinite Sasakian manifold \(\widetilde M\) has a close relation with the geometry of structure vector field \(\xi\), the distributions \(TM^\perp\), and \(\varphi(TM^\perp)\).

MSC:

53B30 Local differential geometry of Lorentz metrics, indefinite metrics
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53B25 Local submanifolds
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