Certain submanifolds of a Kenmotsu manifold (Q2752844)

Let \(L\) be a line and \(F\) a Kählerian manifold. The warped product space \(L\times_fF\) with a differentiable function \(f\) is said to be a Kenmotsu manifold [\textit{K. Kenmotsu}, Tohoku Math. J., II. Ser. 24, 91-103 (1972; Zbl 0245.53040)]. It is a \((2m+1)\)-dimensional Riemannian manifold \(\widetilde M\) which admits an endomorphism \(\varphi\) of its tangent bundle \(T \widetilde M\), a vector field \(\xi\) and a 1-form \(\eta\), satisfying certain conditions. An \((n+1)\)-dimensional submanifold \(M\) tangent to \(\xi\) in \(\widetilde M\) is called a generalized contact CR-submanifold if the maximal anti-invariant subspaces by \(\varphi\) NEWLINE\[NEWLINED^\perp_x= T_xM\cap \varphi(T^\perp_x M),\;x\in MNEWLINE\]NEWLINE in \(T_xM\) define a differentiable subbundle of \(TM\). If here \(\varphi\) is replaced by \(J\), i.e. ``anti-invariant'' is replaced by ``totally real'', \(M\) is said to be a generalized CR-submanifold [\textit{I. Mihai}, Certain submanifolds of a Kähler manifold, Geom. Topology of Submanifolds, World Scientific, 265-268 (1996; Zbl 0936.53039)].NEWLINENEWLINENEWLINENecessary and sufficient conditions are given for a generalized contact CR-submanifold of a Kenmotsu manifold to be a contact CR-submanifold. The same is done for involutivity of the subbundle \(D\oplus \{\xi\}\). For contact CR-submanifolds in a Kenmotsu space form an inequality B. Y. Chen's is established.NEWLINENEWLINEFor the entire collection see [Zbl 0953.00035].