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Semi-slant submanifolds of a Sasakian manifold. (English) Zbl 0944.53028
Let $\stackrel{˜}{M}$ be an almost contact metric manifold and $\left(\varphi ,\xi ,\eta ,g\right)$ its almost contact metric structure [see, e.g., D. E. Blair, Lect. Notes Math. 509, Springer-Verlag (1976; Zbl 0319.53026)]. Let $M$ be a Riemannian manifold isometrically immersed in $\stackrel{˜}{M}$, for which the structure vector field $\xi$ is tangent to $M$. Denote by $𝒟$ the orthogonal complement of $\xi$ in $TM$. For a nonzero vector $X\in {T}_{p}M$, which is not colinear with ${\xi }_{p}$, denote by $\theta \left(X\right)$ the angle between $\varphi X$ and ${𝒟}_{p}$. The submanifold (the distribution $𝒟$) is said to be slant [A. Lotta, Bull. Math. Soc. Roum. 39, 183-198 (1996; Zbl 0885.53058)] if the angle $\theta \left(X\right)$ is independent of the choice of $X\in {𝒟}_{p}$ and $p\in M$. The authors define $M$ to be semi-slant if there exist two orthogonal distributions ${𝒟}_{1}$, ${𝒟}_{2}$ such that $TM={𝒟}_{1}\oplus {𝒟}_{2}\oplus \left\{\xi \right\}$, ${𝒟}_{1}$ is invariant ($\varphi \left({𝒟}_{1}\right)={𝒟}_{1}$) and ${𝒟}_{2}$ is slant. They find necessary and sufficient conditions for $M$ to be semi-slant, and study properties of such submanifolds. Among other things, they study integrability conditions for various distributions involved with the definition of semi-slantness in the case when the ambient manifold is Sasakian. See also recent papers by the same authors.

##### MSC:
 53C40 Global submanifolds (differential geometry) 53D15 Almost contact and almost symplectic manifolds 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)