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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 pN and XT p N the angle θ 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 (θ=0) and totally real (θ=π/2) submanifolds.

Let M n be a submanifold immersed in a semi-Riemannian manifold (M n+k ,g). The distribution Rad(TM)=TMTM 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 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:
53C40Global submanifolds (differential geometry)
53C15Differential geometric structures on manifolds
53C50Lorentz manifolds, manifolds with indefinite metrics