On Two Different Classes of Warped Product Submanifolds of Kenmotsu Manifolds

Shyamal Kumar Hui ; Md. Hasan Shahid ; Tanumoy Pal ; Joydeb Roy

Kragujevac Journal of Mathematics, Tome 47 (2023) no. 6, p. 965

Résumé

Warped product skew CR-submanifold of the form $M=M_1\times_f M_\perp$ of a Kenmotsu manifold $\bar{M}$ (throughout the paper), where $M_1=M_T×M_\theta$ and $M_T,\ M_\perp,\ M_\theta$ represents invariant, anti-invariant and proper slant submanifold of $\bar{M}$, studied in \cite{NAGHI} and another class of warped product skew CR-submanifold of the form $M=M_2\times_fM_T$ of $\bar{M}$, where $M_2=M_\perp×M_\theta$ is studied in \cite{HPR}. Also the warped product submanifold of the form $M=M_3\times_fM_\theta$ of $\bar{M}$, where $M_3=M_T×M_\bot$ and $M_T,\ M_\perp,\ M_\theta$ represents invariant, anti-invariant and proper point wise slant submanifold of $\bar{M}$, were studied in \cite{HSRP}. As a generalization of the above mentioned three classes, we consider a class of warped product submanifold of the form $M=M_4\times_fM_{\theta_3}$ of $\bar{M}$, where $M_4=M_{\theta_1}×M_{\theta_2}$ in which $M_{\theta_1}$ and $M_{\theta_2}$ are proper slant submanifolds of $\bar{M}$ and $M_{\theta_3}$ represents a proper pointwise slant submanifold of $\bar{M}$. A characterization is given on the existence of such warped product submanifolds which generalizes the characterization of warped product submanifolds of the form $M=M_1\times_f M_\perp$, studied in \cite{NAGHI}, the characterization of warped product submanifolds of the form $M=M_2\times_fM_T$, studied in \cite{HPR}, the characterization of warped product submanifolds of the form $M=M_3\times_fM_\theta$, studied in \cite{HSRP} and also the characterization of warped product pointwise bi-slant submanifolds of $\bar{M}$, studied in \cite{HRP}. Since warped product bi-slant submanifolds of $\bar{M}$ does not exist (Theorem 4.2 of \cite{HRP}), the Riemannian product $M_4=M_{\theta_1}×M_{\theta_2}$ cannot be a warped product. So, for studying the bi-warped product submanifolds of $\bar{M}$ of the form $M_{\theta_1}\times_{f_1} M_{\theta_2}\times_{f_2} M_{\theta_3}$, we have taken $M_{\theta_1},\ M_{\theta_2},\ M_{\theta_3}$ as pointwise slant submanifolds of $\bar{M}$ of distinct slant functions $\theta_1,\ \theta_2,\ \theta_3$ respectively. The existence of such type of bi-warped product submanifolds of $\bar{M}$ is ensured by an example. Finally, a Chen-type inequality on the squared norm of the second fundamental form of such bi-warped product submanifolds of $\bar{M}$ is obtained which also generalizes the inequalities obtained in \cite{UDDINB1}, \cite{HSRP} and \cite{HRP}, respectively.

Détail

Classification : 53C15, 53C40