#
Five-Dimensional Contact CR-Submanifolds in
S
7
(
1
)
^{ †}

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

**Theorem**

**1.**

**Remark**

**1.**

## 2. Preliminaries

#### 2.1. Sasakian Manifolds

#### 2.2. Submanifolds

#### 2.3. Contact $CR$-Submanifolds

#### 2.4. $CR$ Warped Product Submanifolds in Sasakian Manifolds

#### 2.5. The Sasakian Structure on ${\mathbb{S}}^{2m+1}(1)$

#### 2.6. Problem

## 3. Five-Dimensional Nearly Totally Geodesic Contact $\mathit{C}\mathit{R}$-Submanifolds in ${\mathbb{S}}^{\mathbf{7}}(\mathbf{1})$

#### 3.1. Essential Characteristics of Five-Dimensional Contact $CR$-Submanifolds in ${\mathbb{S}}^{7}(1)$

**Lemma**

**1.**

- (i)
- if ${a}_{1}+{a}_{3}=0$ and ${b}_{1}+{b}_{3}=0$ then $\mathrm{trace}A(X)=0$ for any $X\in H(M)$;
- (ii)
- if ${a}_{1}+{a}_{3}=0$ and ${b}_{1}+{b}_{3}\ne 0$, then take $t=0$ and denote ${E}_{2}={e}_{1}$ and ${E}_{1}=-{e}_{2}$;
- (iii)
- if ${a}_{1}+{a}_{3}\ne 0$ and ${b}_{1}+{b}_{3}=0$, then take $t=\pi /2$ and denote ${E}_{2}={e}_{2}$ and ${E}_{1}={e}_{1}$;
- (iv)
- if both ${a}_{1}+{a}_{3}\ne 0$ and ${b}_{1}+{b}_{3}\ne 0$, then take t such that $tant=-\frac{{a}_{1}+{a}_{3}}{{b}_{1}+{b}_{3}}$; denote the corresponding X by ${E}_{2}$ and set ${E}_{1}=-\phi {E}_{2}$.

**Proposition**

**1.**

**Proposition**

**2.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Lemma**

**3.**

**Lemma**

**4.**

**Proof.**

**Case 1.**${a}_{1}={a}_{3}$ and

**Case 2.**${a}_{1}\ne {a}_{3}$.

**Case 1.**For the sake of simplicity, we make the following notation ${a}_{1}={a}_{3}:=A$

**ER**) imply new relations between the functions we have considered:

**Case 2.**As ${a}_{1}\ne {a}_{3}$, we immediately obtain $\omega =1$, $p={a}_{1}+{a}_{3}$ and a, c, q, l, r, $\theta $, $\alpha $ and $\beta $ vanish. Moreover, we should have ${a}_{1}{a}_{3}=-1$. Again, for the sake of simplicity we denote ${a}_{1}=A$ and hence ${a}_{3}=-\frac{1}{A}$ and $p=A-\frac{1}{A}$. Obviously, A cannot vanish and satisfies the following partial differential equations

**ER**) imply no new relations.

#### 3.2. The Case 1: M Is Congruent to ${\mathbb{S}}^{3}{\times}_{f}{\mathbb{S}}^{2}$

- (a)
- $\mathsf{P}{\nabla}_{Z}W=g(Z,W){\mathsf{X}}_{\mathsf{0}}$, for all $Z,W$ in ${\mathcal{D}}^{\perp}$;
- (b)
- $\mathsf{P}{\nabla}_{Z}{\mathsf{X}}_{\mathsf{0}}=0$, for all Z in ${\mathcal{D}}^{\perp}$;
- (c)
- $\mathsf{Q}{\nabla}_{X}Y=0$, for all $X,Y$ in $\mathcal{D}$;

- $\mathsf{\Phi}({N}_{1}\times \{{p}_{2}\})$ is an integral manifold for $\mathcal{D}$ for every ${p}_{2}\in {N}_{2}$;
- $\mathsf{\Phi}(\{{p}_{1}\}\times {N}_{2})$ is an integral manifold for ${\mathcal{D}}^{\perp}$ for every ${p}_{1}\in {N}_{1}$;

**Proposition**

**3.**

**Remark**

**2.**

- on one hand $[{E}_{3},{E}_{4}]=\frac{1}{{T}^{2}cosy}\left({T}_{v}{E}_{3}-{T}_{u}{E}_{4}\right)$,
- and on the other hand $[{E}_{3},{E}_{4}]=-\alpha {E}_{3}-\beta {E}_{4}$.

**Remark**

**3.**

**Remark**

**4.**

- $T(u,v)=\frac{2}{1+{u}^{2}+{v}^{2}}$, associated to the parametrization$$(u,v)\mapsto \left(\frac{2u}{1+{u}^{2}+{v}^{2}},\frac{2v}{1+{u}^{2}+{v}^{2}},\frac{{u}^{2}+{v}^{2}-1}{1+{u}^{2}+{v}^{2}}\right)$$
- $T(u,v)=\frac{1}{coshv}$, associated to the parametrization$$(u,v)\mapsto \left(\frac{cosu}{coshv},\frac{sinu}{coshv},tanhv\right)$$

**Remark**

**6.**

**Remark**

**7.**

#### 3.3. The Case 2: M is Congruent to ${\mathbb{S}}^{3}{\times}_{{f}_{1}}{\mathbb{S}}^{1}{\times}_{{f}_{2}}{\mathbb{S}}^{1}$

- from (99) we get$$\left\{\begin{array}{c}{U}_{xz}-\frac{1}{2}U=0,\hfill \\ {V}_{xz}+\frac{1}{2}V=0;\hfill \end{array}\right.$$
- from (96) we obtain$$\left\{\begin{array}{c}{U}_{z}+2{U}_{x}=0,\hfill \\ {V}_{z}-2{V}_{x}=0;\hfill \end{array}\right.$$
- from (101) we have$$\left\{\begin{array}{c}{U}_{xx}+\frac{1}{4}U=0,\hfill \\ {V}_{xx}+\frac{1}{4}V=0;\hfill \end{array}\right.$$
- from (95) we find$$\left\{\begin{array}{c}{U}_{zz}+U=0,\hfill \\ {V}_{zz}+V=0.\hfill \end{array}\right.$$

- (i)
- the decomposition $T(M)={D}_{0}\perp \phantom{\rule{-11.49994pt}{0ex}}\u25ef{D}_{3}\perp \phantom{\rule{-11.49994pt}{0ex}}\u25ef{D}_{4}$ is orthogonal;(here $\perp \phantom{\rule{-11.49994pt}{0ex}}\u25ef$ means the orthogonal decomposition);
- (ii)
- the distributions ${D}_{3}$ and ${D}_{4}$ are spherical;
- (iii)
- the distributions ${D}_{3}^{\perp}$ and ${D}_{4}^{\perp}$ are autoparallel, that is ${\nabla}_{Z}W\in {D}_{k}^{\perp}$, ($k=3,4$), for any $Z,W\in {D}_{k}^{\perp}$.

- $\mathsf{\Phi}({N}_{0}\times \{{p}_{3}\}\times \{{p}_{4}\})$ is an integral manifold for ${D}_{0}$ for every ${p}_{3}\in {N}_{3}$, ${p}_{4}\in {N}_{4}$;
- $\mathsf{\Phi}(\{{p}_{0}\}\times {N}_{3}\times \{{p}_{4}\})$ is an integral manifold for ${D}_{3}$ for every ${p}_{0}\in {N}_{0}$, ${p}_{4}\in {N}_{4}$;
- $\mathsf{\Phi}(\{{p}_{0}\}\times \{{p}_{3}\}\times {N}_{4})$ is an integral manifold for ${D}_{4}$ for every ${p}_{0}\in {N}_{0}$, ${p}_{3}\in {N}_{3}$.

**Proposition**

**4.**

## 4. Conclusions and Further Research

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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The Symmetry We Use | The Result We Get |
---|---|

$(\overline{\nabla}h)({E}_{3},{E}_{3},{E}_{4})$ | $\begin{array}{c}{a}_{1}{b}_{2}=0\hfill \\ {b}_{1}({a}_{1}+{a}_{3})=0\hfill \end{array}$ |

$(\overline{\nabla}h)({E}_{3},{E}_{4},{E}_{4})$ | $\begin{array}{c}{a}_{3}{b}_{2}=0\hfill \\ {b}_{1}({a}_{1}+{a}_{3})=0\hfill \end{array}$ |

$(\overline{\nabla}h)({E}_{1},{E}_{2},{E}_{3})$ | $\begin{array}{c}{E}_{1}({b}_{1})+p{a}_{1}+1-2{b}_{2}r={E}_{2}({a}_{1})+q{b}_{1}-1={a}_{1}^{2}-{b}_{1}^{2}-{b}_{2}^{2}\hfill \\ {E}_{1}({b}_{2})+2{b}_{1}r=l({a}_{1}-{a}_{3})+q{b}_{2}=0\hfill \end{array}$ |

$(\overline{\nabla}h)({E}_{1},{E}_{2},{E}_{4})$ | $\begin{array}{c}-{E}_{1}({b}_{1})+p{a}_{3}+1+2{b}_{2}r={E}_{2}({a}_{3})-q{b}_{1}-1={a}_{3}^{2}-{b}_{1}^{2}-{b}_{2}^{2}\hfill \\ {E}_{1}({b}_{2})+2{b}_{1}r=l({a}_{1}-{a}_{3})+q{b}_{2}=0\hfill \end{array}$ |

The Symmetry We Use | The Result We Get | |
---|---|---|

L1 | $(\overline{\nabla}h)({E}_{3},{E}_{3},{E}_{4})$ | ${b}_{1}({a}_{1}+{a}_{3})=0$ |

L2 | $(\overline{\nabla}h)({E}_{1},{E}_{2},{E}_{3})$ | $\begin{array}{c}{E}_{1}({b}_{1})+p{a}_{1}+1={E}_{2}({a}_{1})+q{b}_{1}-1={a}_{1}^{2}-{b}_{1}^{2}\hfill \\ 2{b}_{1}r=l({a}_{1}-{a}_{3})=0\hfill \end{array}$ |

L3 | $(\overline{\nabla}h)({E}_{1},{E}_{2},{E}_{4})$ | $-{E}_{1}({b}_{1})+p{a}_{3}+1={E}_{2}({a}_{3})-q{b}_{1}-1={a}_{3}^{2}-{b}_{1}^{2}$ |

L4 | $(\overline{\nabla}h)({E}_{1},{E}_{1},{E}_{3})$ | $\begin{array}{c}{E}_{1}({a}_{1})-p{b}_{1}=-2{a}_{1}{b}_{1}\hfill \\ r({a}_{1}-{a}_{3})=0\hfill \end{array}$ |

L5 | $(\overline{\nabla}h)({E}_{1},{E}_{1},{E}_{4})$ | ${E}_{1}({a}_{3})+p{b}_{1}=2{a}_{3}{b}_{1}$ |

L6 | $(\overline{\nabla}h)({E}_{1},{E}_{3},{E}_{3})$ | $\begin{array}{c}{E}_{3}({a}_{1})-a{b}_{1}=0\hfill \\ \alpha ({a}_{1}-{a}_{3})=0\hfill \end{array}$ |

L7 | $(\overline{\nabla}h)({E}_{1},{E}_{4},{E}_{4})$ | $\begin{array}{c}{E}_{4}({a}_{3})+c{b}_{1}=0\hfill \\ \beta ({a}_{1}-{a}_{3})=0\hfill \end{array}$ |

L8 | $(\overline{\nabla}h)({E}_{2},{E}_{2},{E}_{3})$ | $\begin{array}{c}{E}_{2}({b}_{1})-q{a}_{1}=2{a}_{1}{b}_{1}\hfill \\ 2l{b}_{1}=0\hfill \end{array}$ |

L9 | $(\overline{\nabla}h)({E}_{2},{E}_{2},{E}_{4})$ | ${E}_{2}({b}_{1})+q{a}_{3}=2{a}_{3}{b}_{1}$ |

L10 | $(\overline{\nabla}h)({E}_{2},{E}_{3},{E}_{3})$ | $\begin{array}{c}{E}_{3}({b}_{1})+a{a}_{1}=0\hfill \\ 2\alpha {b}_{1}=0\hfill \end{array}$ |

L11 | $(\overline{\nabla}h)({E}_{2},{E}_{4},{E}_{4})$ | $\begin{array}{c}{E}_{4}({b}_{1})-c{a}_{3}=0\hfill \\ 2\beta {b}_{1}=0\hfill \end{array}$ |

L12 | $(\overline{\nabla}h)({E}_{1},{E}_{3},\xi )$ | $\begin{array}{c}\xi ({a}_{1})=(\omega -1){b}_{1}\hfill \\ \theta ({a}_{1}-{a}_{3})=0\hfill \end{array}$ |

L13 | $(\overline{\nabla}h)({E}_{1},{E}_{4},\xi )$ | $\xi ({a}_{3})=(1-\omega ){b}_{1}$ |

L14 | $(\overline{\nabla}h)({E}_{2},{E}_{3},\xi )$ | $\begin{array}{c}\xi ({b}_{1})=(1-\omega ){a}_{1}\hfill \\ 2\theta {b}_{1}=0\hfill \end{array}$ |

L15 | $(\overline{\nabla}h)({E}_{2},{E}_{4},\xi )$ | $\xi ({b}_{1})=(\omega -1){a}_{3}$ |

L16 | $(\overline{\nabla}h)({E}_{1},{E}_{3},{E}_{4})$ | $\begin{array}{c}{E}_{3}({a}_{3})+a{b}_{1}=0\hfill \\ {E}_{4}({a}_{1})-c{b}_{1}=0\hfill \end{array}$ |

L17 | $(\overline{\nabla}h)({E}_{2},{E}_{3},{E}_{4})$ | $\begin{array}{c}{E}_{3}({b}_{1})-a{a}_{3}=0\hfill \\ {E}_{4}({b}_{1})+c{a}_{1}=0\hfill \end{array}$ |

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Djorić, M.; Munteanu, M.I. Five-Dimensional Contact *CR*-Submanifolds in *Mathematics* **2020**, *8*, 1278.
https://doi.org/10.3390/math8081278

**AMA Style**

Djorić M, Munteanu MI. Five-Dimensional Contact *CR*-Submanifolds in *Mathematics*. 2020; 8(8):1278.
https://doi.org/10.3390/math8081278

**Chicago/Turabian Style**

Djorić, Mirjana, and Marian Ioan Munteanu. 2020. "Five-Dimensional Contact *CR*-Submanifolds in *Mathematics* 8, no. 8: 1278.
https://doi.org/10.3390/math8081278