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Slant lightlike submanifolds of indefinite Hermitian manifolds. (English) Zbl 1158.53045

Let \((M,g,J)\) be an almost Hermitian manifold. A submanifold \(N\) of \((M,g,J)\) is called slant if for each \(p\in N\) and \(X\in T_ pN\) the angle \(\theta\) between \(JX\) and \(T_ pN\) is constant [B.-Y. Chen, Bull. Aust. Math. Soc. 41, No. 1, 135–147 (1990; Zbl 0677.53060)]. Special cases of slant submanifolds are almost complex (\(\theta=0\)) and totally real (\(\theta=\pi/2\)) submanifolds.
Let \(M^ n\) be a submanifold immersed in a semi-Riemannian manifold \((M^{n+k},g)\). The distribution \(\text{Rad}\,(TM)=TM\cap TM^\perp\) is called the radical and its complementary distribution \(S(TM)\) is called the screen distribution. A submanifold \(M^ n\) is called a light-like submanifold if \(\text{Rad}\,(TM)\) is of rank \(k\) [K. L. Duggal and A. Bejancu Lightlike submanifolds of semi-Riemannian manifolds and applications (Mathematics and its Applications Dordrecht: Kluwer Academic Publishers) (1996; Zbl 0848.53001)].
The goal of this paper is to introduce the notion of a slant light-like submanifold of an indefinite Hermitian manifold \((M,g,J)\). The author proves a characterization theorem for the existence of slant light-like submanifolds and shows that co-isotropic \(CR\)-light-like submanifolds are slant light-like submanifolds. Also, minimal slant light-like submanifolds are presented and some examples and two characterization theorems are given.

MSC:

53C40 Global submanifolds
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
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