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

Let $\left(M,g,J\right)$ be an almost Hermitian manifold. A submanifold $N$ of $\left(M,g,J\right)$ is called slant if for each $p\in N$ and $X\in {T}_{p}N$ the angle $\theta$ between $JX$ and ${T}_{p}N$ 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 $\left({M}^{n+k},g\right)$. The distribution $\text{Rad}\phantom{\rule{0.166667em}{0ex}}\left(TM\right)=TM\cap T{M}^{\perp }$ is called the radical and its complementary distribution $S\left(TM\right)$ is called the screen distribution. A submanifold ${M}^{n}$ is called a light-like submanifold if $\text{Rad}\phantom{\rule{0.166667em}{0ex}}\left(TM\right)$ 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 $\left(M,g,J\right)$. 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 (differential geometry) 53C15 Differential geometric structures on manifolds 53C50 Lorentz manifolds, manifolds with indefinite metrics