# On a New type of Tensor on Real Hypersurfaces in Non-Flat Complex Space Forms

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

- a complex structure J defined $J:T\tilde{N}\to \tilde{N}$, where $T\tilde{N}$ is the tangent space of $\tilde{N}$, satisfying relations ${J}^{2}=-Id$ and $\tilde{\nabla}J=0$, i.e., J is parallel with respect to the Levi-Civita connection $\tilde{\nabla}$ of $\tilde{N}$
- and a Riemanian metric G that is compatible with J, i.e., $G(JX,JY)=G(X,Y)$ for all tangent X, Y on $\tilde{N}$.

- $\xi =-JN$ is the structure vector field,
- $\varphi $ is a skew-symmetric tensor field of type (1,1) called structure tensor field and defined to be the tangential component of $JX=\varphi X+\eta \left(X\right)N$, for all tangent vectors X to M,
- $\eta $ is a 1-form and is given by the relation $\eta \left(X\right)=g(X,\xi )$ for all tangent vectors X to M,
- g is the induced Riemannian metric on M.

- type (A) which are either geodesic hyperspheres of radius r, $0<r<\frac{\pi}{2}$, or tubes of radius r, with $0<r<\frac{\pi}{2}$ over totally geodesic $\mathbb{C}{P}^{k}$,$1\le k\le n-2$,
- type (B) which are tubes of radius r, $0<r<\frac{\pi}{4}$, over the complex quadric ${Q}^{n-1}$,
- type (C) which are tubes over the Serge embedding of $\mathbb{C}{P}^{1}\times \mathbb{C}{P}^{m}$, with $2m+1=n$ and $n\ge 5$,
- type (D) which are tubes over the Plücker embedding of the Grassmann manifold ${G}_{2,5}$ and $n=9$,

- type (A) which are either horospheres, or geodesic hyperspheres, or tubes over totally geodesic complex hyperbolic hyperplane, or tubes over totally geodesic $\mathbb{C}{H}^{k}$, $1\le k\le n-2$,
- type (B) which are tubes over totally geodesic real hyperbolic space $\mathbb{R}{H}^{2}$ (type (B)).

**Theorem**

**1.**

**Theorem**

**2.**

## 2. Preliminaries

**Theorem**

**3.**

- (i)
- α is constant.
- (ii)
- If W is a vector field which belongs to $\mathbb{D}$ such that $AW=\lambda W$, then$$\begin{array}{c}\hfill (\lambda -\frac{\alpha}{2})A\left(\varphi W\right)=(\frac{\lambda \alpha}{2}+\frac{c}{4})\left(\varphi W\right).\end{array}$$
- (iii)
- If the vector field W satisfies $AW=\lambda W$ and $A\left(\varphi W\right)=\nu \left(\varphi W\right)$ then$$\begin{array}{c}\hfill \lambda \nu =\frac{\alpha}{2}(\lambda +\nu )+\frac{c}{4}.\end{array}$$

**Lemma**

**1.**

**Theorem**

**4.**

## 3. Proof of Theorems 1 and 2

**Case I**: ${\alpha}^{2}+c\ne 0$.

**Case II**: ${\alpha}^{2}+c=0$.

**Remark**

**1.**

- A geodesic hypersphere of radius $r=\frac{\pi}{4}$ in $\mathbb{C}{P}^{n}$ has $\alpha =0$.

**Proposition**

**1.**

## 4. Discussion

- it is worthwhile to study if there are non-Hopf real hypersurfaces of dimension greater than three in non-flat complex space forms with vanishing ${}^{\ast}$-Weyl curvature tensor,
- the ${}^{\ast}$-Weyl curvature tensor could also be defined on real hypersurfaces in other symmetric Hermitian space forms such as the complex two-plane Grassmannians or the complex hyperbolic two-plane Grassmannians and it could be examined if there are real hypersurfaces with vanishing ${}^{\ast}$-Weyl curvature tensor.

## 5. Conclusions

- We introduced a new type of tensor on real hypersurfaces in non-flat complex space forms by defining the ${}^{\ast}$-Weyl curvature tensor on them. The new tensor is related to the ${}^{\ast}$-Ricci tensor of a real hypersurface.
- We initiated the study of real hypersurfaces in non-flat complex space forms in terms of this new tensor. The first geometric condition is that of the vanishing ${}^{\ast}$-Weyl curvature tensor. The motivation for choosing this geometric condition is the existing results for Riemannian manifolds in terms of the Weyl curvature tensor. Thus, we proved two classifications Theorems. The first Theorem concerns Hopf hypersurfaces in non-flat complex space forms of dimension greater or equal to three with vanishing ${}^{\ast}$-Weyl curvature tensor. The second Theorem provides a complete classification for three dimensional real hypersurfaces with vanishing ${}^{\ast}$-Weyl curvature tensor.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Kaimakamis, G.; Panagiotidou, K.
On a New type of Tensor on Real Hypersurfaces in Non-Flat Complex Space Forms. *Symmetry* **2019**, *11*, 559.
https://doi.org/10.3390/sym11040559

**AMA Style**

Kaimakamis G, Panagiotidou K.
On a New type of Tensor on Real Hypersurfaces in Non-Flat Complex Space Forms. *Symmetry*. 2019; 11(4):559.
https://doi.org/10.3390/sym11040559

**Chicago/Turabian Style**

Kaimakamis, George, and Konstantina Panagiotidou.
2019. "On a New type of Tensor on Real Hypersurfaces in Non-Flat Complex Space Forms" *Symmetry* 11, no. 4: 559.
https://doi.org/10.3390/sym11040559