An improved Chen-Ricci inequality for special slant submanifolds in Kenmotsu space forms (Q2872676)

The Chen-Ricci inequality was established by \textit{B. Y. Chen} in [Glasgow Math. J. 41, 33-44 (1999)]. For any \(n\)-dimensional submanifold \(M^{n}\) of a real space form \(\tilde{M}^{n}(c)\) of constant sectional curvature \(c\) holds: NEWLINE\[NEWLINE\text{Ric}(X) \leq (n-1)c+\frac{n^{2}}{4}\| H\| ^{2}.NEWLINE\]NEWLINE It was improved by \textit{H. Deng} in [Int. Electron. J. Geom. 2, No. 2, 39--45 (2009; Zbl 1191.53037)] who showed that for Lagrangian submanifolds \(M^{n}\) in complex space forms \(\tilde{M}^{n}(c)\) holds: NEWLINE\[NEWLINE\text{Ric}(X) \leq (n-1)(c+\frac{n}{4}\| H\|^2).NEWLINE\]NEWLINE Here, the authors develop an inequality for an \((n+1)\)-dimensional special contact \(\theta\)-slant submanifold of a \((2n+1)\)-dimensional Kenmotsu space form \(\tilde{M}^{n}(c)\): NEWLINE\[NEWLINE\text{Ric}(X) \leq \frac{(n+1)^2(n-1)}{4n}\| H\| ^2-1+\frac{(n-1)(c-3)}{4}+\frac{3(c+1)}{4}\cos^{2}\theta.NEWLINE\]NEWLINE The paper treats the equality case as well.