# Casorati Inequalities for Submanifolds in a Riemannian Manifold of Quasi-Constant Curvature with a Semi-Symmetric Metric Connection

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## Abstract

**:**

**2014**, 2014, 327.

## 1. Introduction

## 2. Preliminaries

**Lemma 1.**

## 3. Main Results

**Definition 2.**

**Theorem 3.**

- (i)
- The normalized δ-Casorati curvature ${\delta}_{c}(n-1)$ satisfies$$\rho \le {\delta}_{c}(n-1)+a+\frac{2b}{n}\parallel {U}^{\top}{\parallel}^{2}-\frac{2}{n}\lambda $$$${A}_{n+1}=\left(\begin{array}{cccccc}a& 0& 0& \cdots & 0& 0\\ 0& a& 0& \cdots & 0& 0\\ 0& 0& a& \cdots & 0& 0\\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0& 0& 0& \cdots & a& 0\\ 0& 0& 0& \cdots & 0& 2a\end{array}\right),\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{A}_{n+2}=\cdots ={A}_{n+p}=0$$
- (ii)
- The normalized δ-Casorati curvature ${\widehat{\delta}}_{c}(n-1)$ satisfies$$\rho \le {\widehat{\delta}}_{c}(n-1)+a+\frac{2b}{n}\parallel {U}^{\top}{\parallel}^{2}-\frac{2}{n}\lambda $$$${A}_{n+1}=\left(\begin{array}{cccccc}2a& 0& 0& \cdots & 0& 0\\ 0& 2a& 0& \cdots & 0& 0\\ 0& 0& 2a& \cdots & 0& 0\\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0& 0& 0& \cdots & 2a& 0\\ 0& 0& 0& \cdots & 0& a\end{array}\right),\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{A}_{n+2}=\cdots ={A}_{n+p}=0$$

**Remark 1.**

**Lemma 4.**

**Proof.**

**Lemma 5.**

**Proof of Theorem 3**

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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Zhang, P.; Zhang, L.
Casorati Inequalities for Submanifolds in a Riemannian Manifold of Quasi-Constant Curvature with a Semi-Symmetric Metric Connection. *Symmetry* **2016**, *8*, 19.
https://doi.org/10.3390/sym8040019

**AMA Style**

Zhang P, Zhang L.
Casorati Inequalities for Submanifolds in a Riemannian Manifold of Quasi-Constant Curvature with a Semi-Symmetric Metric Connection. *Symmetry*. 2016; 8(4):19.
https://doi.org/10.3390/sym8040019

**Chicago/Turabian Style**

Zhang, Pan, and Liang Zhang.
2016. "Casorati Inequalities for Submanifolds in a Riemannian Manifold of Quasi-Constant Curvature with a Semi-Symmetric Metric Connection" *Symmetry* 8, no. 4: 19.
https://doi.org/10.3390/sym8040019