# Chen Inequalities for Warped Product Pointwise Bi-Slant Submanifolds of Complex Space Forms and Its Applications

^{*}

## Abstract

**:**

## 1. Introduction

**Theorem**

**1.**

**Theorem**

**2.**

**Theorem**

**3.**

## 2. Preliminaries and Notations

**Definition**

**1.**

**Definition**

**2.**

- (i)
- $T{M}^{n}={\mathcal{D}}_{{\theta}_{1}}\oplus {\mathcal{D}}_{{\theta}_{2}};$
- (ii)
- $J{\mathcal{D}}_{{\theta}_{1}}\perp {\mathcal{D}}_{{\theta}_{2}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}and\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}J{\mathcal{D}}_{{\theta}_{2}}\perp {\mathcal{D}}_{{\theta}_{1}};$
- (iii)
- Each distribution ${\mathcal{D}}_{{\theta}_{i}}$ is a pointwise slant with a slant function ${\theta}_{i}:{T}^{*}M\to \mathbb{R}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}for\phantom{\rule{0.166667em}{0ex}}i=1,2.$

**Remark**

**1.**

**Remark**

**2.**

**Remark**

**3.**

**Remark**

**4.**

**Remark**

**5.**

## 3. Main Inequality for Warped Product Pointwise Bi-Slant Submanifolds

**Lemma**

**1.**

**Proof**

**of**

**Theorem**

**2.**

#### Consequences of Theorem 2

**Corollary**

**1.**

**Corollary**

**2.**

**Corollary**

**3.**

**Corollary**

**4.**

**Corollary**

**5.**

**Proof**

**of**

**Theorem**

**3.**

**Corollary**

**6.**

**Corollary**

**7.**

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Ali, A.; Alkhaldi, A.H.
Chen Inequalities for Warped Product Pointwise Bi-Slant Submanifolds of Complex Space Forms and Its Applications. *Symmetry* **2019**, *11*, 200.
https://doi.org/10.3390/sym11020200

**AMA Style**

Ali A, Alkhaldi AH.
Chen Inequalities for Warped Product Pointwise Bi-Slant Submanifolds of Complex Space Forms and Its Applications. *Symmetry*. 2019; 11(2):200.
https://doi.org/10.3390/sym11020200

**Chicago/Turabian Style**

Ali, Akram, and Ali H. Alkhaldi.
2019. "Chen Inequalities for Warped Product Pointwise Bi-Slant Submanifolds of Complex Space Forms and Its Applications" *Symmetry* 11, no. 2: 200.
https://doi.org/10.3390/sym11020200