## 1. Introduction

## 2. Preliminaries

#### 2.1. Contact Riemannian Structures

**Definition**

**1**

#### 2.2. Strongly Pseudo-Convex almost CR Structures

**Definition**

**2**

- each fiber ${\mathcal{H}}_{p}$, $p\in M$, is of complex dimension n
- $\mathcal{H}\cap \overline{\mathcal{H}}=\left\{0\right\}$, where $\overline{\mathcal{H}}$ denotes the complex conjugation of $\mathcal{H},$
- $[\mathcal{H},\mathcal{H}]\subset \mathcal{H}\phantom{\rule{4pt}{0ex}}(\mathrm{integrability})$.

**Definition**

**3**

## 3. The pseudo-Hermitian Symmetric Spaces

**Definition**

**4**

**Proposition**

**1**

- $(i)$
- $\widehat{\nabla}\eta =0$, $\widehat{\nabla}\xi =0$;
- $(ii)$
- $\widehat{\nabla}g=0$;
- $(iii$-$1)$
- $\widehat{T}(X,Y)=2L(X,JY)\xi $, $X,\phantom{\rule{4pt}{0ex}}Y\in \Gamma (D)$;
- $(iii$-$2)$
- $\widehat{T}(\xi ,\varphi Y)=-\varphi \widehat{T}(\xi ,Y)$, $Y\in \Gamma (D)$;
- $(iv)$
- $({\widehat{\nabla}}_{X}\varphi )Y=\Omega (X,Y)$, $X,\phantom{\rule{4pt}{0ex}}Y\in \Gamma (TM)$.

**Proposition**

**2**

**Corollary**

**1**

**Proposition**

**3**

**Definition**

**5**

**Proposition**

**4**

**Proposition**

**5**

- such a class is invariant under pseudo-homothetic transformations,
- the associated pseudo-Hermitian structure is CR-integrable.

**Theorem**

**1**

**Proposition**

**6.**

**Theorem**

**2.**

**Proof of Theorem**

**2.**

**Corollary**

**2.**

## 4. The Contact Strongly Pseudo-Convex 3-Manifolds

**Theorem**

**3.**

**Proof of Theorem**

**3.**

**Corollary**

**3.**

**Corollary**

**4.**

**Proposition**

**7**

**Theorem**

**4.**

**Proof of Theorem**

**4.**

**Example**

**1.**

**Example**

**2.**

- $f=z$: ${M}_{f}$ has a constant holomorphic sectional curvature $\widehat{H}=3$. It is not locally ϕ-symmetric in the sense of Reference [5].
- $f=\alpha $ (const.): ${M}_{f}$ has a constant holomorphic sectional curvature $\widehat{H}=2+{\alpha}^{2}$. It is curvature homogeneous. A locally ϕ-symmetric space occurs only when $\alpha =0$.
- $f=\alpha tan(\alpha arctanz)$, $\alpha >0$ (const.) and if ${M}_{f}$ is restricted to the set $\{(x,y,z)\in {\mathbb{R}}^{3}|-\frac{\pi}{2\alpha}<arctanz<\frac{\pi}{2\alpha}\}$: it has a constant holomorphic sectional curvature $\widehat{H}=2-{\alpha}^{2}$. It is not locally ϕ-symmetric.

**Remark**

**1.**

## Funding

## Conflicts of Interest

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