On generic submanifolds of manifolds endowed with metric mixed 3-structures (Q2828652)

The author introduces the concept of generic submanifold \(M\) as a non-degenerate submanifold of \(\tilde{M}\) with a metric mixed 3-structure \(((\varphi_{\alpha},\xi_{\alpha},\eta_{\alpha})_{\alpha=1,2,3},\bar{g})\) such that the structure vector fields \(\xi_{1},\xi_{2},\xi_{3}\) are tangent to \(M\) and \(\varphi_{\alpha}(T_{p}^{\perp})\subset T_{p}M\) for all \(p\in M\) and \(\alpha=1,2,3\). From this definition, there exist three distributions \(\mathcal{D}_{\alpha}\) defined as \(\mathcal{D}_{\alpha}(p)=\varphi_{\alpha}(T_{p}^{\perp})\) for \(p\in M\), \(\alpha=1,2,3\). Moreover, there exist three 1-dimensional distributions spanned by the three structure vector fields \(\xi_{\alpha}\). In this paper, the author investigates these canonical distributions and finds several properties as below. NEWLINENEWLINEFirst, every \(\mathcal{D}_{\alpha}\) and \(\xi_{\alpha}\) are all mutually orthogonal to each other. Therefore, \(\mathcal{D}^{\perp}=\mathcal{D}_{1}\oplus\mathcal{D}_{2}\oplus\mathcal{D}_{3}\) and \(\xi={\xi_{1}}\oplus {\xi_{2}} \oplus {\xi_{3}}\) are orthogonal. So, we have the decomposition \(TM=\mathcal{D}\oplus \mathcal{D}^{\perp}\oplus \xi\). NEWLINENEWLINESecondly, suppose that \(M\) is a generic submanifold of a mixed 3-cosymplectic or mixed 3-Sasakian manifold \(\tilde{M}\). Then several results are shown as below. NEWLINENEWLINE\((a)\) The distributions \(\mathcal{D}_{\alpha}\) are integrable.NEWLINENEWLINE\((b)\) The distribution \(\mathcal{D}^{\perp}\oplus \xi\) is integrable if and only if \(M\) is a \((\mathcal{D},\mathcal{D}^{\perp})\)-geodesic submanifold. NEWLINENEWLINE\((c)\) If the distribution \(\mathcal{D}\oplus \xi\) is integrable, then each leaf of \(\mathcal{D}\oplus \xi\) is totally geodesic immersed in the ambient manifold \(\tilde{M}\). NEWLINENEWLINE\((d)\) If \(M\) is a \(\mathcal{D}\)-geodesic submanifold, then \(M\) is ruled.NEWLINENEWLINEBesides these, more details are included.