Semi-slant submanifolds of a Sasakian manifold (Q1818587)

Let \(\widetilde M\) be an almost contact metric manifold and \((\varphi,\xi,\eta,g)\) its almost contact metric structure [see, e.g., \textit{D. E. Blair}, Lect. Notes Math. 509, Springer-Verlag (1976; Zbl 0319.53026)]. Let \(M\) be a Riemannian manifold isometrically immersed in \(\widetilde M\), for which the structure vector field \(\xi\) is tangent to \(M\). Denote by \(\mathcal D\) the orthogonal complement of \(\xi\) in \(TM\). For a nonzero vector \(X\in T_pM\), which is not colinear with \(\xi_p\), denote by \(\theta(X)\) the angle between \(\varphi X\) and \(\mathcal D_p\). The submanifold (the distribution \(\mathcal D\)) is said to be slant [\textit{A. Lotta}, Bull. Math. Soc. Roum. 39, 183-198 (1996; Zbl 0885.53058)] if the angle \(\theta(X)\) is independent of the choice of \(X\in\mathcal D_p\) and \(p\in M\). The authors define \(M\) to be semi-slant if there exist two orthogonal distributions \(\mathcal D_1\), \(\mathcal D_2\) such that \(TM=\mathcal D_1\oplus\mathcal D_2\oplus\{\xi\}\), \(\mathcal D_1\) is invariant (\(\varphi(\mathcal D_1)=\mathcal D_1\)) and \(\mathcal D_2\) 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.