# A Comprehensive Survey on Parallel Submanifolds in Riemannian and Pseudo-Riemannian Manifolds

## Abstract

**:**

Table of Contents | |

Section 1. Introduction | . . . . . . . . . . . . . . . . . . . . . . 3 |

Section 2. Preliminaries | . . . . . . . . . . . . . . . . . . . . . . 4 |

2.1. Basic definitions, formulas and equations | . . . . . . . . . . . . . . . . . . . . . . 4 |

2.2. Indefinite real space forms | . . . . . . . . . . . . . . . . . . . . . . 5 |

2.3. Gauss image | . . . . . . . . . . . . . . . . . . . . . . 6 |

Section 3. Some general properties of parallel submanifolds | . . . . . . . . . . . . . . . . . . . . . . 7 |

Section 4. Parallel submanifolds of Euclidean spaces | . . . . . . . . . . . . . . . . . . . . . . 7 |

4.1. Gauss map and parallel submanifolds | . . . . . . . . . . . . . . . . . . . . . . 7 |

4.2. Normal sections and parallel submanifolds | . . . . . . . . . . . . . . . . . . . . . . 8 |

4.3. Symmetric submanifolds and parallel submanifolds | . . . . . . . . . . . . . . . . . . . . . . 9 |

4.4. Extrinsic k-symmetric submanifolds as ${\nabla}^{c}$-parallel submanifolds | . . . . . . . . . . . . . . . . . . . . . . 9 |

Section 5. Symmetric R-spaces and parallel submanifolds of real space forms | . . . . . . . . . . . . . . . . . . . . . . 10 |

5.1. Symmetric R-spaces and Borel subgroups | . . . . . . . . . . . . . . . . . . . . . . 10 |

5.2. Classification of symmetric R-spaces | . . . . . . . . . . . . . . . . . . . . . . 10 |

5.3. Ferus’ theorem | . . . . . . . . . . . . . . . . . . . . . . 11 |

5.4. Parallel submanifolds in spheres | . . . . . . . . . . . . . . . . . . . . . . 11 |

5.5. Parallel submanifolds in hyperbolic spaces | . . . . . . . . . . . . . . . . . . . . . . 11 |

Section 6. Parallel Kaehler submanifolds | . . . . . . . . . . . . . . . . . . . . . . 11 |

6.1. Segre and Veronese maps | . . . . . . . . . . . . . . . . . . . . . . 12 |

6.2. Classification of parallel Kaehler submanifolds of $C{P}^{m}$ and $C{H}^{m}$ | . . . . . . . . . . . . . . . . . . . . . . 12 |

6.3. Parallel Kaehler submanifolds of Hermitian symmetric spaces | . . . . . . . . . . . . . . . . . . . . . . 13 |

6.4. Parallel Kaehler manifolds in complex Grassmannian manifolds | . . . . . . . . . . . . . . . . . . . . . . 13 |

Section 7. Parallel totally real submanifolds | . . . . . . . . . . . . . . . . . . . . . . 14 |

7.1. Basics on totally real submanifolds | . . . . . . . . . . . . . . . . . . . . . . 14 |

7.2. Parallel Lagrangian submanifolds in $C{P}^{m}$ | . . . . . . . . . . . . . . . . . . . . . . 14 |

7.3. Parallel surfaces of $C{P}^{2}$ and $C{H}^{2}$ | . . . . . . . . . . . . . . . . . . . . . . 15 |

7.4. Parallel totally real submanifolds in nearly Kaehler ${S}^{6}$ | . . . . . . . . . . . . . . . . . . . . . . 16 |

Section 8. Parallel slant submanifolds of complex space forms | . . . . . . . . . . . . . . . . . . . . . . 16 |

8.1. Basics on slant submanifolds | . . . . . . . . . . . . . . . . . . . . . . 16 |

8.2. Classification of parallel slant submanifolds | . . . . . . . . . . . . . . . . . . . . . . 17 |

Section 9. Parallel submanifolds of quaternionic space forms and Cayley plane | . . . . . . . . . . . . . . . . . . . . . . 17 |

9.1. Parallel submanifolds of quaternionic space forms | . . . . . . . . . . . . . . . . . . . . . . 17 |

9.2. Parallel submanifolds of the Cayley plane | . . . . . . . . . . . . . . . . . . . . . . 18 |

Section 10. Parallel spatial submanifolds in pseudo-Euclidean spaces | . . . . . . . . . . . . . . . . . . . . . . 18 |

10.1. Marginally trapped surfaces | . . . . . . . . . . . . . . . . . . . . . . 18 |

10.2. Classification of parallel spatial surfaces in ${\mathbb{E}}_{s}^{m}$ | . . . . . . . . . . . . . . . . . . . . . . 18 |

10.3. Special case: parallel spatial surfaces in ${\mathbb{E}}_{1}^{3}$ | . . . . . . . . . . . . . . . . . . . . . . 19 |

Section 11. Parallel spatial surfaces in ${S}_{s}^{m}$ | . . . . . . . . . . . . . . . . . . . . . . 19 |

11.1. Classification of parallel spatial surfaces in ${S}_{s}^{m}$ | . . . . . . . . . . . . . . . . . . . . . . 19 |

11.2. Special case: parallel spatial surfaces in ${S}_{1}^{3}$ | . . . . . . . . . . . . . . . . . . . . . . 21 |

Section 12. Parallel spatial surfaces in ${H}_{s}^{m}$ | . . . . . . . . . . . . . . . . . . . . . . 22 |

12.1. Classification of parallel spatial surfaces in ${H}_{s}^{m}$ | . . . . . . . . . . . . . . . . . . . . . . 22 |

12.2. A parallel spatial surfaces in ${H}_{2}^{4}$ | . . . . . . . . . . . . . . . . . . . . . . 23 |

12.3. Special case: parallel surfaces in ${H}_{1}^{3}$ | . . . . . . . . . . . . . . . . . . . . . . 24 |

Section 13. Parallel Lorentz surfaces in pseudo-Euclidean spaces | . . . . . . . . . . . . . . . . . . . . . . 24 |

13.1. Classification of parallel Lorentzian surfaces in ${\mathbb{E}}_{s}^{m}$ | . . . . . . . . . . . . . . . . . . . . . . 25 |

13.2. Classification of parallel Lorentzian surfaces in ${E}_{1}^{3}$ | . . . . . . . . . . . . . . . . . . . . . . 26 |

Section 14. Parallel surfaces in a light cone $\mathcal{L}C$ | . . . . . . . . . . . . . . . . . . . . . . 26 |

14.1. Light cones in general relativity | . . . . . . . . . . . . . . . . . . . . . . 26 |

14.2. Parallel surfaces in ${\mathcal{L}C}_{1}^{3}\subset {\mathbb{E}}_{1}^{4}$ | . . . . . . . . . . . . . . . . . . . . . . 27 |

14.3. Parallel surfaces in ${\mathcal{L}C}_{2}^{3}\subset {\mathbb{E}}_{2}^{4}$ | . . . . . . . . . . . . . . . . . . . . . . 27 |

Section 15. Parallel surfaces in de Sitter space-time ${S}_{1}^{4}$ | . . . . . . . . . . . . . . . . . . . . . . 27 |

15.1. Classification of parallel spatial surfaces in de Sitter space-time ${S}_{1}^{4}$ | . . . . . . . . . . . . . . . . . . . . . . 28 |

15.2. Classification of parallel Lorentzian surfaces in de Sitter space-time ${S}_{1}^{4}$ | . . . . . . . . . . . . . . . . . . . . . . 29 |

Section 16. Parallel surfaces in anti-de Sitter space-time ${H}_{1}^{4}$ | . . . . . . . . . . . . . . . . . . . . . . 29 |

16.1. Classification of parallel spatial surfaces in ${H}_{1}^{4}$ | . . . . . . . . . . . . . . . . . . . . . . 29 |

16.2. Classification of parallel Lorentzian surfaces in anti-de Sitter space-time ${H}_{1}^{4}$ | . . . . . . . . . . . . . . . . . . . . . . 30 |

16.3. Special case: parallel Lorentzian surfaces in ${H}_{1}^{3}$ | . . . . . . . . . . . . . . . . . . . . . . 31 |

Section 17. Parallel spatial surfaces in ${S}_{2}^{4}$ | . . . . . . . . . . . . . . . . . . . . . . 31 |

17.1. Four-dimensional manifolds with neutral metrics | . . . . . . . . . . . . . . . . . . . . . . 31 |

17.2. Classification of parallel Lorentzian surfaces in ${S}_{2}^{4}$ | . . . . . . . . . . . . . . . . . . . . . . 32 |

17.3. Classification of parallel Lorentzian surfaces in ${H}_{2}^{4}$ | . . . . . . . . . . . . . . . . . . . . . . 33 |

Section 18. Parallel spatial surfaces in ${S}_{3}^{4}$ and in ${H}_{3}^{4}$ | . . . . . . . . . . . . . . . . . . . . . . 34 |

18.1. Classification of parallel spatial surfaces in ${S}_{3}^{4}$ | . . . . . . . . . . . . . . . . . . . . . . 34 |

18.2. Classification of parallel spatial surfaces in ${H}_{3}^{4}$ | . . . . . . . . . . . . . . . . . . . . . . 34 |

Section 19. Parallel Lorentzian surfaces in ${\mathbb{C}}^{n}$, $C{P}_{1}^{2}$ and $C{H}_{1}^{2}$ | . . . . . . . . . . . . . . . . . . . . . . 35 |

19.1. Hopf fibration | . . . . . . . . . . . . . . . . . . . . . . 35 |

19.2. Classification of parallel spatial surfaces in ${\mathbb{C}}_{1}^{2}$ | . . . . . . . . . . . . . . . . . . . . . . 36 |

19.3. Classification of parallel Lorentzian surface in $C{P}_{1}^{2}$ | . . . . . . . . . . . . . . . . . . . . . . 36 |

19.4. Classification of parallel Lorentzian surface in $C{H}_{1}^{2}$ | . . . . . . . . . . . . . . . . . . . . . . 38 |

Section 20. Parallel Lorentz surfaces in $I{\times}_{f}{R}^{n}\left(c\right)$ | . . . . . . . . . . . . . . . . . . . . . . 38 |

20.1. Basics on Robertson–Walker space-times | . . . . . . . . . . . . . . . . . . . . . . 38 |

20.2. Parallel submanifolds of Robertson-Walker space-times | . . . . . . . . . . . . . . . . . . . . . . 39 |

Section 21. Thurston’s eight 3-dimensional model geometries | . . . . . . . . . . . . . . . . . . . . . . 39 |

Section 22. Parallel surfaces in three-dimensional Lie groups | . . . . . . . . . . . . . . . . . . . . . . 40 |

22.1. Milnor’s classification of 3-dimensional unimodular Lie groups | . . . . . . . . . . . . . . . . . . . . . . 40 |

22.2. Parallel surfaces in the motion group $E(1,1)$ | . . . . . . . . . . . . . . . . . . . . . . 41 |

22.3. Parallel surfaces in $So{l}_{3}$ | . . . . . . . . . . . . . . . . . . . . . . 41 |

22.4. Parallel surfaces in the motion group $E\left(2\right)$ | . . . . . . . . . . . . . . . . . . . . . . 42 |

22.5. Parallel surfaces in $SU\left(2\right)$ | . . . . . . . . . . . . . . . . . . . . . . 42 |

22.6. Parallel surfaces in the real special linear group $SL(2,\mathbb{R})$ | . . . . . . . . . . . . . . . . . . . . . . 43 |

22.7. Parallel surfaces in non-unimodular three-dimensional Lie groups | . . . . . . . . . . . . . . . . . . . . . . 44 |

22.8. Parallel surfaces in the Heisenberg group $Ni{l}_{3}$ | . . . . . . . . . . . . . . . . . . . . . . 45 |

Section 23. Parallel surfaces in three-dimensional Lorentzian Lie groups | . . . . . . . . . . . . . . . . . . . . . . 45 |

23.1. Three-dimensional Lorentzian Lie groups | . . . . . . . . . . . . . . . . . . . . . . 46 |

23.2. Classification of parallel surfaces in three-dimensional Lorentzian Lie groups | . . . . . . . . . . . . . . . . . . . . . . 47 |

Section 24. Parallel surfaces in reducible three-spaces | . . . . . . . . . . . . . . . . . . . . . . 49 |

24.1. Classification of parallel surfaces in reducible three-spaces | . . . . . . . . . . . . . . . . . . . . . . 49 |

24.2. Parallel surfaces in Walker three-manifolds | . . . . . . . . . . . . . . . . . . . . . . 50 |

Section 25. Bianchi–Cartan–Vranceasu spaces | . . . . . . . . . . . . . . . . . . . . . . 50 |

25.1. Basics on Bianchi–Cartan–Vranceasu spaces | . . . . . . . . . . . . . . . . . . . . . . 50 |

25.2. B-scrolls | . . . . . . . . . . . . . . . . . . . . . . 51 |

25.3. Parallel surfaces in Bianchi–Cartan–Vranceasu spaces | . . . . . . . . . . . . . . . . . . . . . . 51 |

Section 26. Parallel surfaces in homogeneous three-spaces | . . . . . . . . . . . . . . . . . . . . . . 52 |

26.1. Homogeneous three-spaces | . . . . . . . . . . . . . . . . . . . . . . 52 |

26.2. Classification of parallel surfaces in homogeneous Lorentzian three-spaces | . . . . . . . . . . . . . . . . . . . . . . 52 |

Section 27. Parallel surfaces in Lorentzian symmetric three-spaces | . . . . . . . . . . . . . . . . . . . . . . 52 |

27.1. Lorentzian symmetric three-spaces | . . . . . . . . . . . . . . . . . . . . . . 53 |

27.2. Classification of parallel surfaces in homogeneous Lorentzian three-spaces | . . . . . . . . . . . . . . . . . . . . . . 54 |

Section 28. Three natural extensions of parallel submanifolds | . . . . . . . . . . . . . . . . . . . . . . 55 |

28.1. Submanifolds with parallel mean curvature vector | . . . . . . . . . . . . . . . . . . . . . . 55 |

28.2. Higher order parallel submanifolds | . . . . . . . . . . . . . . . . . . . . . . 56 |

28.3. Semi-parallel submanifolds | . . . . . . . . . . . . . . . . . . . . . . 56 |

References | . . . . . . . . . . . . . . . . . . . . . . 57–64 |

## 1. Introduction

## 2. Preliminaries

#### 2.1. Basic Definitions, Formulas and Equations

#### 2.2. Indefinite Real Space Forms

**Example**

**1.**

#### 2.3. Gauss Image

**Theorem**

**1.**

## 3. Some General Properties of Parallel Submanifolds

**Definition**

**1.**

**Lemma**

**1.**

**Lemma**

**2.**

**Lemma**

**3.**

## 4. Parallel Submanifolds of Euclidean Spaces

#### 4.1. Gauss Map and Parallel Submanifolds

**Theorem**

**2.**

- (i)
- If the relative nullity $\nu =0$, then M is a complete totally geodesic submanifold of $G(n,m-n)$.
- (ii)
- If $\nu \ge 1$, then there exists a $(G\left({\mathbb{E}}^{m}\right),{\mathbb{E}}^{m})$-fibration $\pi :M\to B$, where B is a complete totally geodesic submanifold of $G(n,m-n)$ and the fibres are the leaves of the relative nullity foliation. The metric of M is composed from those on base and fibre and the fibration admits an integrable connection with totally geodesic horizontal leaves (i.e., it is a totally geodesic Riemannian submersion).
- (iii)
- The original Riemannian connection of M or its projection onto B, respectively, coincides with the connection induced from $G(n,m-n)$.
- (iv)
- M has nonnegative curvature and is locally symmetric.

**Theorem**

**3.**

- (a)
- A surface in an affine 3-space ${\mathbb{E}}^{3}$ of ${\mathbb{E}}^{m}$.
- (b)
- A surface of $\phantom{\rule{0.166667em}{0ex}}{\mathbb{E}}^{m}$ with parallel second fundamental form, that is, M is a parallel surface.
- (c)
- A surface in an affine 4-space ${\mathbb{E}}^{4}$ of ${\mathbb{E}}^{m}$ which is locally the Riemannian product of two plane curves of non-zero curvature.
- (d)
- A complex curve lying fully in ${\mathbb{C}}^{2}$, where ${\mathbb{C}}^{2}$ denotes an affine ${\mathbb{E}}^{4}$ endowed with some orthogonal almost complex structure.

**Theorem**

**4.**

**Theorem**

**5.**

- (a)
- a real hypersurface or
- (b)
- a parallel submanifold or
- (c)
- a complex hypersurface.

#### 4.2. Normal Sections and Parallel Submanifolds

**Theorem**

**6.**

**Theorem**

**7.**

**Theorem**

**8.**

#### 4.3. Symmetric Submanifolds and Parallel Submanifolds

**Theorem**

**9.**

#### 4.4. Extrinsic K-Symmetric Submanifolds as ${\nabla}^{c}$-Parallel Submanifolds

**Theorem**

**10.**

- (1)
- M admits a canonical connection ${\nabla}^{c}$ such that ${\nabla}^{c}h=0$,
- (2)
- M is an extrinsic homogeneous submanifold with constant principal curvatures,
- (3)
- M is an orbit of an s-representation, that is, of an isotropy representation of a semisimple Riemannian symmetric space.

## 5. Symmetric R-Spaces and Parallel Submanifolds of Real Space Forms

#### 5.1. Symmetric R-Spaces and Borel Subgroups

#### 5.2. Classification of Symmetric R-Spaces

- (a)
- all Hermitian symmetric spaces of compact type,
- (b)
- Grassmann manifolds $O(p+q)/O\left(p\right)\times O\left(q\right),Sp(p+q)/Sp\left(p\right)\times Sp\left(q\right),$
- (c)
- the classical groups $SO\left(m\right),\phantom{\rule{0.166667em}{0ex}}U\left(m\right),\phantom{\rule{0.166667em}{0ex}}Sp\left(m\right)$,
- (d)
- $U\left(2m\right)/Sp\left(m\right),\phantom{\rule{0.166667em}{0ex}}U\left(m\right)/O\left(m\right)$,
- (e)
- $\left(SO\right(p+1)\times SO(q+1\left)\right)/S\left(O\right(p)\times O(q\left)\right)$, where $S\left(O\right(p)\phantom{\rule{-0.166667em}{0ex}}\times \phantom{\rule{-0.166667em}{0ex}}O(q\left)\right)$ is the subgroup of $SO(p+1)\times SO(q+1)$ consisting of matrices of the form$$\left(\begin{array}{cccc}\epsilon & 0& & \\ 0& A& & \\ & & \epsilon & 0\\ & & 0\hfill & \hfill B\end{array}\right),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\epsilon =\pm 1,\phantom{\rule{1.em}{0ex}}A\in O\left(p\right),\phantom{\rule{1.em}{0ex}}B\in O\left(q\right),$$
- (f)
- the Cayley projective plane $\mathcal{O}{P}^{2}$ and
- (g)
- the three exceptional spaces ${E}_{6}/Spin\left(10\right)\times T,{E}_{7}/{E}_{6}\times T,$ and ${E}_{6}/{F}_{4}.$

#### 5.3. Ferus’ Theorem

**Theorem**

**11.**

- (1)
- $M={\mathbb{E}}^{{m}_{0}}\times {M}_{1}\times \cdots \times {M}_{s}\subset {\mathbb{E}}^{{m}_{0}}\times {\mathbb{E}}^{{m}_{1}}\times \cdots \times {\mathbb{E}}^{{m}_{s}}={\mathbb{E}}^{m}$, $s\ge 0$, or to
- (2)
- $M={M}_{1}\times \cdots \times {M}_{s}\subset {\mathbb{E}}^{{m}_{1}}\times \cdots \times {\mathbb{E}}^{{m}_{s}}={\mathbb{E}}^{m}$, $s\ge 1$,

#### 5.4. Parallel Submanifolds in Spheres

#### 5.5. Parallel Submanifolds in Hyperbolic Spaces

**Theorem**

**12.**

- (1)
- If M is not contained in any complete totally geodesic hypersurface of ${H}^{m}\left(\overline{c}\right)$, then M is congruent to the product$${H}^{{m}_{0}}\left({c}_{0}\right)\times {M}_{1}\times \cdots \times {M}_{s}\subset {H}^{{m}_{0}}\left({c}_{0}\right)\times {S}^{m-{m}_{0}-1}\left({c}^{\prime}\right)\subset {H}^{{m}_{0}}\left(\overline{c}\right)$$
- (2)
- If M is contained in a complete totally geodesic hypersurface N of ${H}^{m}\left(\overline{c}\right)$, then N is isometric to an $(m-1)$-sphere or to a Euclidean $(m-1)$-space or to a hyperbolic $(m-1)$-space. Consequently, such parallel submanifolds reduce to the parallel submanifolds described before.

## 6. Parallel Kaehler Submanifolds

#### 6.1. The Segre and Veronese Maps

**Theorem**

**13.**

**Theorem**

**14.**

#### 6.2. Classification of Parallel Kaehler Submanifolds of $C{P}^{m}$ and $C{H}^{m}$

**Theorem**

**15.**

**Theorem**

**16.**

**Theorem**

**17.**

#### 6.3. Parallel Kaehler Submanifolds of Hermitian Symmetric Spaces

**Theorem**

**18.**

#### 6.4. Parallel Kaehler Manifolds in Complex Grassmannian Manifolds

**Theorem**

**19.**

## 7. Parallel Totally Real Submanifolds

#### 7.1. Basics on Totally Real Submanifolds

**Theorem**

**20.**

**Theorem**

**21.**

**Remark**

**1.**

**Remark**

**2.**

#### 7.2. Parallel Lagrangian Submanifolds of $C{P}^{n}$

**Theorem**

**22.**

- (1)
- M is locally the Calabi product of a point with a lower-dimensional parallel Lagrangian submanifold;
- (2)
- M is locally the Calabi product of two lower-dimensional parallel Lagrangian submanifolds; or
- (3)
- M is congruent to one of the following symmetric spaces: (a) $SU\left(k\right)/SO\left(k\right)$ with $n=k(k+1)/2-1$ and $k\ge 3$, (b) $SU\left(k\right)$ with $n={k}^{2}-1$ and $k\ge 3$, $SU\left(2k\right)/Sp\left(k\right)$ with $n=2{k}^{2}-k-1$ and $k\ge 3$ or (c) ${E}_{6}/{F}_{4}$ with $n=26$.

#### 7.3. Parallel Surfaces of $C{P}^{2}$ and $C{H}^{2}$

**Theorem**

**23.**

- (a)
- If M is holomorphic, then locally either
- (a.1)
- M is a totally geodesic complex projective line $C{P}^{1}\left(4\right)$ in $C{P}^{2}\left(4\right)$ or
- (a.2)
- M is the complex quadric ${Q}^{1}$ embedded in $C{P}^{2}\left(4\right)$ as $\left\{({z}_{0},{z}_{1},{z}_{2})\in C{P}^{2}\left(4\right)\phantom{\rule{4pt}{0ex}}|\phantom{\rule{4pt}{0ex}}{z}_{0}^{2}+{z}_{1}^{2}+{z}_{2}^{2}=0\right\},$ where ${z}_{0},{z}_{1},{z}_{2}$ are complex homogeneous coordinates on $C{P}^{2}\left(4\right)$.

- (b)
- If M is Lagrangian, then locally either
- (b.1)
- M is a totally geodesic real projective plane $R{P}^{2}\left(1\right)$ in $C{P}^{2}\left(4\right)$ or
- (b.2)
- M is a flat surface and the immersion is congruent to $\pi \circ L$, where $\pi :{S}^{5}\left(1\right)\to C{P}^{2}\left(4\right)$ is the Hopf-fibration and $L:M\to {S}^{5}\left(1\right)\subseteq {\mathbb{C}}^{3}$ is given by$$\begin{array}{c}\phantom{\rule{36.135pt}{0ex}}L(x,y)=(\frac{a\phantom{\rule{0.166667em}{0ex}}{e}^{-ix/a}}{\sqrt{1+{a}^{2}}},\frac{{e}^{i(ax+by)}}{\sqrt{1+{a}^{2}+{b}^{2}}}sin\left(\sqrt{1+{a}^{2}+{b}^{2}}\phantom{\rule{0.166667em}{0ex}}y\right),\hfill \\ \hfill \phantom{\rule{21.68121pt}{0ex}}\frac{{e}^{i(ax+by)}}{\sqrt{1+{a}^{2}}}\left(cos\left(\sqrt{1+{a}^{2}+{b}^{2}}\phantom{\rule{0.166667em}{0ex}}y\right)-\frac{ib}{\sqrt{1+{a}^{2}+{b}^{2}}}sin\left(\sqrt{1+{a}^{2}+{b}^{2}}\phantom{\rule{0.166667em}{0ex}}y\right)\right)),\end{array}$$

**Theorem**

**24.**

- (a)
- If ${M}^{2}$ is holomorphic, then it is an open part of a totally geodesic complex submanifold $C{H}^{1}(-4)$ in $C{H}^{2}(-4)$.
- (b)
- If M is Lagrangian, then locally either
- (b.1)
- M is a totally geodesic real hyperbolic plane $R{H}^{2}(-1)$ in $C{H}^{2}(-4)$ or
- (b.2)
- M is flat and the immersion is congruent to $\pi \circ L$, where $\pi :{H}_{1}^{5}(-1)\to C{H}^{2}(-4)$ is the Hopf fibration and $L:{M}^{2}\to {H}_{1}^{5}(-1)\subseteq {\mathbb{C}}_{1}^{3}$ is one of the following six maps:
- (1)
- $L=(\frac{{e}^{i(ax+by)}}{\sqrt{1-{a}^{2}}}\left(cosh\left(\sqrt{1-{a}^{2}-{b}^{2}}\phantom{\rule{0.166667em}{0ex}}y\right)-\frac{ib\phantom{\rule{0.166667em}{0ex}}sinh\left(\sqrt{1-{a}^{2}-{b}^{2}}\phantom{\rule{0.166667em}{0ex}}y\right)}{\sqrt{1-{a}^{2}-{b}^{2}}}\right),$$$\frac{{e}^{i(ax+by)}}{\sqrt{1-{a}^{2}-{b}^{2}}}sinh\left(\sqrt{1-{a}^{2}-{b}^{2}}\phantom{\rule{0.166667em}{0ex}}y\right),\frac{a\phantom{\rule{0.166667em}{0ex}}{e}^{ix/a}}{\sqrt{1-{a}^{2}}}),\phantom{\rule{0.277778em}{0ex}}a,b\in \mathbf{R},\phantom{\rule{0.277778em}{0ex}}a\ne 0,\phantom{\rule{0.277778em}{0ex}}{a}^{2}+{b}^{2}<1;$$
- (2)
- $L(x,y)=\left(\phantom{\rule{-0.166667em}{0ex}}\left(\frac{i}{b}+y\right){e}^{i(\sqrt{1-{b}^{2}}x+by)},y{e}^{i(\sqrt{1-{b}^{2}}x+by)},\frac{\sqrt{1-{b}^{2}}}{b}{e}^{ix/\sqrt{1-{b}^{2}}}\right),$$\phantom{\rule{0.277778em}{0ex}}b\in \mathbf{R},\phantom{\rule{4pt}{0ex}}0<{b}^{2}<1;$
- (3)
- $L(x,y)=(\frac{{e}^{i(ax+by)}}{\sqrt{1-{a}^{2}}}\left(cos\left(\sqrt{{a}^{2}+{b}^{2}-1}\phantom{\rule{0.166667em}{0ex}}y\right)-\frac{ib\phantom{\rule{0.166667em}{0ex}}sin\left(\sqrt{{a}^{2}+{b}^{2}-1}\phantom{\rule{0.166667em}{0ex}}y\right)}{\sqrt{{a}^{2}+{b}^{2}-1}}\right),$$$\frac{{e}^{i(ax+by)}}{\sqrt{{a}^{2}+{b}^{2}-1}}sin\left(\sqrt{{a}^{2}+{b}^{2}-1}\phantom{\rule{0.166667em}{0ex}}y),\frac{a\phantom{\rule{0.166667em}{0ex}}{e}^{ix/a}}{\sqrt{1-{a}^{2}}}\right),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}a,b\in \mathbf{R},\phantom{\rule{0.277778em}{0ex}}0<{a}^{2}<1,\phantom{\rule{0.277778em}{0ex}}{a}^{2}+{b}^{2}>1;$$
- (4)
- $L(x,y)=\left(\frac{a\phantom{\rule{0.166667em}{0ex}}{e}^{ix/a}}{\sqrt{{a}^{2}-1}},\frac{{e}^{i(ax+by)}}{\sqrt{{a}^{2}+{b}^{2}-1}}sin\left(\sqrt{{a}^{2}+{b}^{2}-1}\phantom{\rule{0.166667em}{0ex}}y\right),\right.$$$\left.\frac{{e}^{i(ax+by)}}{\sqrt{{a}^{2}-1}}\left(cos\left(\sqrt{{a}^{2}+{b}^{2}-1}\phantom{\rule{0.166667em}{0ex}}y\right)-\frac{ib\phantom{\rule{0.166667em}{0ex}}sin\left(\sqrt{{a}^{2}+{b}^{2}-1}\phantom{\rule{0.166667em}{0ex}}y\right)}{\sqrt{{a}^{2}+{b}^{2}-1}}\right)\right),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}a,b\in \mathbf{R},\phantom{\rule{4pt}{0ex}}{a}^{2}>1;$$
- (5)
- $L(x,y)=\frac{{e}^{ix}}{8{b}^{2}}\left(i+8{b}^{2}(i+x)-4by,i+8{b}^{2}x-4by,4b{e}^{2iby}\right)$, $\phantom{\rule{0.277778em}{0ex}}\mathbf{R}\ni b\ne 0;$
- (6)
- $L(x,y)={e}^{ix}\left(1+\frac{{y}^{2}}{2}-ix,y,\frac{{y}^{2}}{2}-ix\right).$

#### 7.4. Parallel Totally Real Submanifolds in Nearly Kaehler ${S}^{6}$

**Theorem**

**25.**

## 8. Parallel Slant Submanifolds of Complex Space Forms

#### 8.1. Basics on Slant Submanifolds

**Definition**

**2.**

**Theorem**

**26.**

**Theorem**

**27.**

#### 8.2. Classification of Parallel Slant Submanifolds

**Theorem**

**28.**

- (a)
- An open portion of a slant plane in ${\mathbb{C}}^{2}\subset {\mathbb{C}}^{m}$;
- (b)
- An open portion of the product surface of two plane circles;
- (c)
- An open portion of a circular cylinder which is contained in a hyperplane of ${\mathbb{C}}^{2}\subset {\mathbb{C}}^{m}$.

**Theorem**

**29.**

## 9. Parallel Submanifolds of Quaternionic Space Forms and Cayley Plane

#### 9.1. Parallel Submanifolds of Quaternionic Space Forms

#### 9.2. Parallel Submanifolds of the Cayley Plane

## 10. Parallel Spatial Submanifolds in Pseudo-Euclidean Spaces

**Lemma**

**4.**

#### 10.1. Marginally Trapped Surfaces

#### 10.2. Classification of Parallel Spatial Surfaces in ${\mathbb{E}}_{s}^{m}$

**Theorem**

**30.**

- (A)
- the surface is an open part of one of the following 11 surfaces:
- (i)
- a totally geodesic Euclidean 2-plane ${\mathbb{E}}^{2}\subset {\mathbb{E}}_{s}^{m}$ given by $(0,\dots ,0,u,v);$
- (ii)
- a totally umbilical ${S}^{2}\left(1\right)$ in a totally geodesic ${\mathbb{E}}^{3}$ given by $\left(0,\dots ,0,cosu,sinucosv,sinusinv\right);$
- (iii)
- a flat cylinder ${\mathbb{E}}^{1}\times {S}^{1}$ lying in a totally geodesic ${\mathbb{E}}^{3}\subset {\mathbb{E}}_{s}^{m}$ given by $\left(0,\dots ,0,u,cosv,sinv\right);$
- (iv)
- a flat torus ${S}^{1}\times {S}^{1}$ in a totally geodesic ${\mathbb{E}}^{4}$ given by $\left(0,\dots ,0,acosu,asinu,bcosv,bsinv\right)$ with $a,b>0;$
- (v)
- a real projective plane of curvature $\frac{1}{3}$ lying in a totally geodesic ${\mathbb{E}}^{5}\subset {\mathbb{E}}_{s}^{m}$ given by$$\begin{array}{c}\phantom{\rule{14.45377pt}{0ex}}\left(0,\dots ,0,\frac{vw}{\sqrt{3}},\frac{uw}{\sqrt{3}},\frac{uv}{\sqrt{3}},\frac{{u}^{2}-{v}^{2}}{2\sqrt{3}},\frac{1}{6}\left({u}^{2}+{v}^{2}-2{w}^{2}\right)\phantom{\rule{-1.4457pt}{0ex}}\right),\phantom{\rule{0.277778em}{0ex}}{u}^{2}+{v}^{2}+{w}^{2}=3;\hfill \end{array}$$
- (vi)
- a hyperbolic 2-plane ${H}^{2}$ in a totally geodesic ${\mathbb{E}}_{1}^{3}$ as $\left(coshu,0,\dots ,0,sinhucosv,sinhusinv\right);$
- (vii)
- a flat cylinder ${H}^{1}\times {\mathbb{E}}^{1}$ lying in a totally geodesic ${\mathbb{E}}_{1}^{3}\subset {\mathbb{E}}_{1}^{4}$ given by $\left(coshu,0,\dots ,0,sinhu,v\right);$
- (viii)
- a flat surface ${H}^{1}\times {S}^{1}$ in a totally geodesic ${\mathbb{E}}_{1}^{4}\subset {\mathbb{E}}_{s}^{m}$ given by$$\left(acoshu,0,\dots ,0,asinhu,bcosv,bsinv\right)$$
- (ix)
- a flat totally umbilical surface of a totally geodesic ${\mathbb{E}}_{1}^{4}\subset {\mathbb{E}}_{s}^{m}$ defined by$$\left({u}^{2}+{v}^{2}+\frac{1}{4},0,\dots ,0,u,v,{u}^{2}+{v}^{2}-\frac{1}{4}\right);$$
- (x)
- a flat surface ${H}^{1}\times {H}^{1}$ lying in a totally geodesic ${\mathbb{E}}_{2}^{4}\subset {\mathbb{E}}_{s}^{m}$ given by$$\left(acoshu,bcoshv,0,\dots ,0,asinhu,bsinhv\right),\phantom{\rule{0.277778em}{0ex}}a,b>0;$$
- (xi)
- a surface of curvature $-\frac{1}{3}$ lying in a totally geodesic ${\mathbb{E}}_{3}^{5}\subset {\mathbb{E}}_{s}^{m}$ given by$$\begin{array}{c}(sinh\left(\frac{2s}{\sqrt{3}}\right)-\frac{{t}^{2}}{3}-\left(\frac{7}{8}+\frac{{t}^{4}}{18}\right){e}^{\frac{2s}{\sqrt{3}}},\phantom{\rule{0.166667em}{0ex}}t+\left(\frac{{t}^{3}}{3}-\frac{t}{4}\right){e}^{\frac{2s}{\sqrt{3}}},\frac{1}{2}+\frac{{t}^{2}}{2}{e}^{\frac{2s}{\sqrt{3}}},\hfill \\ \phantom{\rule{25.29494pt}{0ex}}0,\dots ,0,t+\left(\frac{{t}^{3}}{3}+\frac{t}{4}\right){e}^{\frac{2s}{\sqrt{3}}},\phantom{\rule{0.166667em}{0ex}}sinh\left(\frac{2s}{\sqrt{3}}\right)-\frac{{t}^{2}}{3}-\left(\frac{1}{8}+\frac{{t}^{4}}{18}\right){e}^{\frac{2s}{\sqrt{3}}}),\phantom{\rule{0.277778em}{0ex}}or\hfill \end{array}$$

- (B)
- $L=({f}_{1},\dots ,{f}_{\ell},\varphi ,{f}_{\ell},\dots ,{f}_{1})$, where ϕ is a surface given by (i), (iii), (iv), (vii), (viii), (ix), or (x) from $\left(A\right)$ and ${f}_{1},\dots ,{f}_{\ell}$ are polynomials of degree $\le 2$ in $u,v$.

#### 10.3. Special Case: Parallel Spatial Surfaces in ${\mathbb{E}}_{1}^{3}$

**Corollary**

**1.**

- (1)
- the Euclidean plane ${\mathbb{E}}^{2}$ given by $(0,u,v)$;
- (2)
- a hyperbolic plane ${H}^{2}$ given by $a(coshucoshv,coshusinhv,sinhu)\phantom{\rule{0.166667em}{0ex}}a>0$;
- (3)
- a cylinder ${H}^{1}\times {\mathbb{E}}^{1}$ defined by $(acoshu,asinhu,v),\phantom{\rule{0.166667em}{0ex}}a>0$;

**Remark**

**3.**

## 11. Parallel Spatial Surfaces in ${S}_{s}^{m}$

#### 11.1. Classification of Parallel Spatial Surfaces in ${S}_{s}^{m}$

**Theorem**

**31.**

- (A)
- the surface is congruent to an open part of one of the following 18 surfaces:
- (1)
- a totally geodesic 2-sphere ${S}^{2}\left(1\right)\subset {S}_{s}^{m}\left(1\right)$;
- (2)
- a totally umbilical ${S}^{2}$ immersed in ${S}_{s}^{m}\left(1\right)\subset {\mathbb{E}}_{s}^{m+1}$ as$$\left(0,\dots ,0,rsinu,rcosucosv,rcosusinv,\sqrt{1-{r}^{2}}\right),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}0<r<1;$$
- (3)
- a totally umbilical ${S}^{2}$ immersed in ${S}_{s}^{m}\left(1\right)\subset {\mathbb{E}}_{s}^{m+1}$ as$$\left(\sqrt{{r}^{2}-1},0,\dots ,0,rsinu,rcosucosv,rcosusinv\right),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}r>1\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}s\ge 1;$$
- (4)
- a flat torus ${S}^{1}\times {S}^{1}$ immersed in ${S}_{s}^{m}\left(1\right)\subset {\mathbb{E}}_{s}^{m+1}$ as$$\left(0,\dots ,0,bcosu,bsinu,ccosv,csinv,\sqrt{1-{b}^{2}-{c}^{2}}\right),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}b,c>0,\phantom{\rule{0.277778em}{0ex}}{b}^{2}+{c}^{2}\le 1;$$
- (5)
- a flat torus ${S}^{1}\times {S}^{1}$ immersed in ${S}_{s}^{m}\left(1\right)\subset {\mathbb{E}}_{s}^{m+1}$ as$$\left(\sqrt{{b}^{2}+{c}^{2}-1},0,\dots ,0,bcosu,bsinu,ccosv,csinv\right),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}b,c,s>0,\phantom{\rule{0.277778em}{0ex}}{b}^{2}+{c}^{2}>1;$$
- (6)
- a real projective plane $R{P}^{2}$ immersed in ${S}_{s}^{m}\left(1\right)\subset {\mathbb{E}}_{s}^{m+1}$ as$$\begin{array}{c}\left(0,\dots ,0,\frac{rvw}{\sqrt{3}},\frac{ruw}{\sqrt{3}},\frac{ruv}{\sqrt{3}},\frac{r({u}^{2}-{v}^{2})}{2\sqrt{3}},\frac{r}{6}({u}^{2}+{v}^{2}-2{w}^{2}),\sqrt{1-{r}^{2}}\right)\hfill \end{array}$$
- (7)
- a real projective plane $R{P}^{2}$ immersed in ${S}_{s}^{m}\left(1\right)\subset {\mathbb{E}}_{s}^{m+1}$ as$$\begin{array}{c}\left(\phantom{\rule{-1.4457pt}{0ex}}\sqrt{{r}^{2}-1},0,\dots ,0,\frac{rvw}{\sqrt{3}},\frac{ruw}{\sqrt{3}},\frac{ruv}{\sqrt{3}},\frac{r({u}^{2}-{v}^{2})}{2\sqrt{3}},\frac{r}{6}({u}^{2}+{v}^{2}-2{w}^{2})\phantom{\rule{-1.4457pt}{0ex}}\right)\hfill \end{array}$$
- (8)
- a hyperbolic 2-plane ${H}^{2}$ immersed in ${S}_{s}^{m}\left(1\right)\subset {\mathbb{E}}_{s}^{m+1}$ as$$\left(rcoshu,0,\dots ,0,rsinhucosv,rsinhusinv,\sqrt{1+{r}^{2}}\right),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}r,s>0;$$
- (9)
- a flat surface ${H}^{1}\times {H}^{1}$ immersed in ${S}_{s}^{m}\left(1\right)\subset {\mathbb{E}}_{s}^{m+1}$ as$$\left(bcoshu,ccoshv,0,\dots ,0,bsinhu,csinhv,\sqrt{1+{b}^{2}+{c}^{2}}\right),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}b,c>0,\phantom{\rule{0.277778em}{0ex}}s\ge 2;$$
- (10)
- a flat surface ${H}^{1}\times {S}^{1}$ immersed in ${S}_{s}^{m}\left(1\right)\subset {\mathbb{E}}_{s}^{m+1}$ as$$\left(bcoshu,0,\dots ,0,bsinhu,ccosv,csinv,\sqrt{1+{b}^{2}-{c}^{2}})\right),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}b,c,s>0,\phantom{\rule{0.277778em}{0ex}}{c}^{2}\le 1+{b}^{2};$$
- (11)
- a flat surface ${H}^{1}\times {S}^{1}$ immersed in ${S}_{s}^{m}\left(1\right)\subset {\mathbb{E}}_{s}^{m+1}$ as$$\left(\sqrt{{c}^{2}-{b}^{2}-1},bcoshu,0,\dots ,0,bsinhu,ccosv,csinv\right),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{c}^{2}>1+{b}^{2}>1;$$
- (12)
- a flat surface immersed in ${S}_{s}^{m}\left(1\right)\subset {\mathbb{E}}_{s}^{m+1}$ as$$r\left({u}^{2}+{v}^{2}+b+\frac{1}{4},0,\dots ,0,\frac{\sqrt{1+b{r}^{2}}}{r},u,v,{u}^{2}+{v}^{2}+b-\frac{1}{4}\right),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}r,s>0,\phantom{\rule{0.277778em}{0ex}}b\ge -{r}^{-2};$$
- (13)
- a flat surface immersed in ${S}_{s}^{m}\left(1\right)\subset {\mathbb{E}}_{s}^{m+1}$ as$$r\left({u}^{2}+{v}^{2}-b+\frac{1}{4},\frac{\sqrt{b{r}^{2}-1}}{r},0,\dots ,0,u,v,{u}^{2}+{v}^{2}-b-\frac{1}{4}\right)$$
- (14)
- a flat surface immersed in ${S}_{s}^{m}\left(1\right)\subset {\mathbb{E}}_{s}^{m+1}$ as$$\begin{array}{c}r\left({u}^{2}+b-\frac{3}{4},0,\dots ,0,\frac{\sqrt{1-(1-b+{c}^{2}){r}^{2}}}{r},u,ccosv,csinv,{u}^{2}+b-\frac{5}{4}\right)\hfill \end{array}$$
- (15)
- a flat surface immersed in ${S}_{s}^{m}\left(1\right)\subset {\mathbb{E}}_{s}^{m+1}$ as$$\begin{array}{c}r\left({u}^{2}+b-\frac{3}{4},\frac{\sqrt{(1-b+{c}^{2}){r}^{2}-1}}{r},0,\dots ,0,u,ccosv,csinv,{u}^{2}+b-\frac{5}{4}\right)\hfill \end{array}$$
- (16)
- a flat surface immersed in ${S}_{s}^{m}\left(1\right)\subset {\mathbb{E}}_{s}^{m+1}$ as$$r\left({v}^{2}-b+\frac{5}{4},ccoshu,0,\dots ,0,\frac{\sqrt{1\phantom{\rule{-1.4457pt}{0ex}}+\phantom{\rule{-1.4457pt}{0ex}}(1\phantom{\rule{-1.4457pt}{0ex}}-\phantom{\rule{-1.4457pt}{0ex}}b\phantom{\rule{-1.4457pt}{0ex}}+\phantom{\rule{-1.4457pt}{0ex}}{c}^{2}){r}^{2}}}{r},csinhu,v,{v}^{2}-b+\frac{3}{4}\right)$$
- (17)
- a flat surface immersed in ${S}_{s}^{m}\left(1\right)\subset {\mathbb{E}}_{s}^{m+1}$ as$$r\left({v}^{2}-b+\frac{5}{4},ccoshu,\frac{\sqrt{(b\phantom{\rule{-1.4457pt}{0ex}}-\phantom{\rule{-1.4457pt}{0ex}}{c}^{2}\phantom{\rule{-1.4457pt}{0ex}}-\phantom{\rule{-1.4457pt}{0ex}}1){r}^{2}\phantom{\rule{-1.4457pt}{0ex}}-\phantom{\rule{-1.4457pt}{0ex}}1}}{r},0,\dots ,0,csinhu,v,{v}^{2}-b+\frac{3}{4}\right)$$
- (18)
- a surface of constant negative curvature immersed in ${S}_{s}^{m}\left(1\right)\subset {\mathbb{E}}_{s}^{m+1}$ as$$\begin{array}{c}\phantom{\rule{21.68121pt}{0ex}}r\left(sinh\left(\frac{2s}{\sqrt{3}}\right)-\frac{{t}^{2}}{3}-\left(\frac{7}{8}+\frac{{t}^{4}}{18}\right){e}^{\frac{2s}{\sqrt{3}}},\phantom{\rule{0.166667em}{0ex}}t+\left(\phantom{\rule{4.pt}{0ex}}\frac{{t}^{3}}{3}-\frac{t}{4}\right){e}^{\frac{2s}{\sqrt{3}}},\phantom{\rule{4.pt}{0ex}}\frac{1}{2}+\frac{{t}^{2}}{2}{e}^{\frac{2s}{\sqrt{3}}},\right.\hfill \\ \hfill \left.\phantom{\rule{36.135pt}{0ex}}0,\dots ,0,t+\left(\frac{{t}^{3}}{3}+\frac{t}{4}\right){e}^{\frac{2s}{\sqrt{3}}},\phantom{\rule{0.166667em}{0ex}}sinh\left(\frac{2s}{\sqrt{3}}\right)-\frac{{t}^{2}}{3}-\left(\frac{1}{8}+\frac{{t}^{4}}{18}\right){e}^{\frac{2s}{\sqrt{3}}},\frac{\sqrt{1+{r}^{2}}}{r}\right)\end{array}$$

- (B)
- $L=({f}_{1},\dots ,{f}_{\ell},\varphi ,{f}_{\ell},\dots ,{f}_{1})$, where ϕ is a surface given by (4), (5) or (9)–(17) from $\left(A\right)$ and ${f}_{1},\dots ,{f}_{\ell}$ are polynomials of degree $\le 2$ in $u,v$ or
- (C)
- $L=(r,\varphi ,r)$, where $r\in {\mathbb{R}}^{+}$ and ϕ is a surface given by (1), (2), (3), (6), (7), (8) or (18) from $\left(A\right)$.

#### 11.2. Special Case: Parallel Spatial Surfaces in ${S}_{1}^{3}$

**Corollary**

**2.**

- (1)
- a totally umbilical sphere ${S}^{2}$ given locally by $(a,bsinu,bcosucosv,bcosusinv),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{b}^{2}-{a}^{2}=1;$
- (2)
- a totally umbilical hyperbolic plane ${H}^{2}$ given by $(acoshucoshv,acoshusinhv,asinhu,b)$ with ${b}^{2}-{a}^{2}=1;$
- (3)
- a flat surface ${H}^{1}\times {S}^{1}$ given by $(acoshu,asinhu,bcosv,bsinv)$ with ${a}^{2}+{b}^{2}=1.$
- (4)
- a totally umbilical Euclidean ${\mathbb{E}}^{2}$ plane given by$$\frac{1}{\sqrt{c}}\left({u}^{2}+{v}^{2}-\frac{3}{4},{u}^{2}+{v}^{2}-\frac{5}{4},u,v\right);$$

**Remark**

**4.**

## 12. Parallel Spatial Surfaces in ${H}_{s}^{m}$

#### 12.1. Classification of Parallel Spatial Surfaces in ${H}_{s}^{m}$

**Theorem**

**32.**

- (A)
- the surface is congruent to an open part of one of the following 18 surfaces:
- (1)
- a totally geodesic ${H}^{2}(-1)$ immersed in ${H}_{s}^{m}(-1)$ as $\left(coshu,0,\dots ,0,sinhucosv,sinhusinv\right)$ with $b>0;$
- (2)
- a totally umbilical ${H}^{2}$ immersed in ${H}_{s}^{m}(-1)\subset {\mathbb{E}}_{s+1}^{m+1}$ as$$\left(rcoshu,0,\dots ,0,rsinhucosv,rsinhusinv,\sqrt{{r}^{2}-1}\phantom{\rule{0.166667em}{0ex}}\right)\phantom{\rule{0.277778em}{0ex}}r>1;$$
- (3)
- a totally umbilical ${H}^{2}$ immersed in ${H}_{s}^{m}(-1)\subset {\mathbb{E}}_{s+1}^{m+1}$ as$$\phantom{\rule{2.em}{0ex}}\phantom{\rule{1.em}{0ex}}\left(rcoshu,\sqrt{1-{r}^{2}},0,\dots ,0,rsinhucosv,rsinhusinv\phantom{\rule{0.166667em}{0ex}}\right),\phantom{\rule{0.277778em}{0ex}}s\ge 1,\phantom{\rule{0.277778em}{0ex}}0<r<1;$$
- (4)
- a totally umbilical ${S}^{2}$ immersed in ${H}_{s}^{m}(-1)\subset {\mathbb{E}}_{s+1}^{m+1}$ as$$\left(\sqrt{1+{r}^{2}},0,\dots ,0,rsinu,rcosucosv,rcosusinv\right),\phantom{\rule{0.277778em}{0ex}}r>0;$$
- (5)
- a flat torus ${S}^{1}\times {S}^{1}$ in ${H}_{s}^{m}(-1)\subset {\mathbb{E}}_{s+1}^{m+1}$ as $(\sqrt{1+{b}^{2}+{c}^{2}}\phantom{\rule{0.166667em}{0ex}},0,\dots ,0,bcosu,bsinu,ccosv,$$csinv,),$ with $b,c>0;$
- (6)
- a surface of constant positive curvature immersed in ${H}_{s}^{m}(-1)\subset {\mathbb{E}}_{s+1}^{m+1}$ as$$\begin{array}{c}\phantom{\rule{36.135pt}{0ex}}\left(\phantom{\rule{-1.4457pt}{0ex}}\sqrt{1+{r}^{2}},0,\dots ,0,\frac{rvw}{\sqrt{3}},\frac{ruw}{\sqrt{3}},\frac{ruv}{\sqrt{3}},\frac{r({u}^{2}-{v}^{2})}{2\sqrt{3}},\frac{r}{6}({u}^{2}+{v}^{2}-2{w}^{2})\phantom{\rule{-1.4457pt}{0ex}}\right)\hfill \end{array}$$
- (7)
- a flat surface ${H}^{1}\times {H}^{1}$ in ${H}_{s}^{m}(-1)$ as $\left(bcoshu,ccoshv,0,\dots ,0,bsinhu,csinhv,\sqrt{{b}^{2}+{c}^{2}-1}\right)$ with $b,c,s>0$ and ${b}^{2}+{c}^{2}\ge 1;$
- (8)
- a flat surface ${H}^{1}\times {H}^{1}$ in ${H}_{s}^{m}(-1)$ as $\left(\sqrt{1-{b}^{2}-{c}^{2}}\phantom{\rule{0.166667em}{0ex}},bcoshu,ccoshv,0,\dots ,0,bsinhu,csinhv\right)$ with $b,c>0$, $s\ge 2$ and ${b}^{2}+{c}^{2}<1;$
- (9)
- a flat surface ${H}^{1}\times {S}^{1}$ in ${H}_{s}^{m}(-1)\subset {\mathbb{E}}_{s+1}^{m+1}$ as $(bcoshu,0,\dots ,0,bsinhu,ccosv,csinv,$$\sqrt{{b}^{2}-{c}^{2}-1})$ with $b,c>0$ and ${b}^{2}\ge {c}^{2}+1$;
- (10)
- a flat surface ${H}^{1}\times {S}^{1}$ immersed in ${H}_{s}^{m}(-1)$ as $(\sqrt{1-{b}^{2}+{c}^{2}}\phantom{\rule{0.166667em}{0ex}},bcoshu,0,\dots ,0,bsinhu,ccosv,$$csinv)$ with $b,c,s>0$ and ${b}^{2}<{c}^{2}+1$;
- (11)
- a flat surface immersed in ${H}_{s}^{m}(-1)\subset {\mathbb{E}}_{s+1}^{m+1}$ as$$r\left({u}^{2}+{v}^{2}+b+\frac{1}{4},0,\dots ,0,\frac{\sqrt{b{r}^{2}-1}}{r},u,v,{u}^{2}+{v}^{2}+b-\frac{1}{4}\phantom{\rule{0.166667em}{0ex}}\right),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}r>1,\phantom{\rule{0.277778em}{0ex}}b\ge {r}^{-2};$$
- (12)
- a flat surface immersed in ${H}_{s}^{m}(-1)\subset {\mathbb{E}}_{s+1}^{m+1}$ as$$r\left({u}^{2}+{v}^{2}-b+\frac{1}{4},\frac{\sqrt{b{r}^{2}+1}}{r},0,\dots ,0,u,v,{u}^{2}+{v}^{2}-b-\frac{1}{4}\phantom{\rule{0.166667em}{0ex}}\right),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}r,s>0,\phantom{\rule{0.277778em}{0ex}}b\ge -{r}^{-2};$$
- (13)
- a flat surface immersed in ${H}_{s}^{m}(-1)\subset {\mathbb{E}}_{s+1}^{m+1}$ as$$\begin{array}{c}r\left({u}^{2}+b-\frac{3}{4},0,\dots ,0,\frac{\sqrt{(b\phantom{\rule{-1.4457pt}{0ex}}-\phantom{\rule{-1.4457pt}{0ex}}{c}^{2}\phantom{\rule{-1.4457pt}{0ex}}-\phantom{\rule{-1.4457pt}{0ex}}1){r}^{2}\phantom{\rule{-1.4457pt}{0ex}}-\phantom{\rule{-1.4457pt}{0ex}}1}}{r},ccosv,csinv,u,{u}^{2}+b-\frac{5}{4}\right)\hfill \end{array}$$
- (14)
- a flat surface immersed in ${H}_{s}^{m}(-1)\subset {\mathbb{E}}_{s+1}^{m+1}$ as$$\begin{array}{c}r\left({u}^{2}+b-\frac{3}{4},\frac{\sqrt{1\phantom{\rule{-1.4457pt}{0ex}}+\phantom{\rule{-1.4457pt}{0ex}}(1\phantom{\rule{-1.4457pt}{0ex}}-\phantom{\rule{-1.4457pt}{0ex}}b\phantom{\rule{-1.4457pt}{0ex}}+\phantom{\rule{-1.4457pt}{0ex}}{c}^{2}){r}^{2}}}{r},0,\dots ,0,ccosv,csinv,u,{u}^{2}+b-\frac{5}{4}\right)\hfill \end{array}$$
- (15)
- a flat surface immersed in ${H}_{s}^{m}(-1)\subset {\mathbb{E}}_{s+1}^{m+1}$ as$$r\left({v}^{2}+b+\frac{5}{4},bcoshu,0,\dots ,0,\frac{\sqrt{(1\phantom{\rule{-1.4457pt}{0ex}}+\phantom{\rule{-1.4457pt}{0ex}}b\phantom{\rule{-1.4457pt}{0ex}}+\phantom{\rule{-1.4457pt}{0ex}}{c}^{2}){r}^{2}\phantom{\rule{-1.4457pt}{0ex}}-\phantom{\rule{-1.4457pt}{0ex}}1}}{r},bsinhu,v,{v}^{2}+b+\frac{3}{4}\right)$$
- (16)
- a flat surface immersed in ${H}_{s}^{m}(-1)\subset {\mathbb{E}}_{s+1}^{m+1}$ as$$r\left({v}^{2}+b+\frac{5}{4},bcoshu,\frac{\sqrt{1\phantom{\rule{-1.4457pt}{0ex}}-\phantom{\rule{-1.4457pt}{0ex}}(a\phantom{\rule{-1.4457pt}{0ex}}+\phantom{\rule{-1.4457pt}{0ex}}b\phantom{\rule{-1.4457pt}{0ex}}+\phantom{\rule{-1.4457pt}{0ex}}{c}^{2}){r}^{2}}}{r},0,\dots ,0,bsinhu,v,{v}^{2}+b+\frac{3}{4}\right)$$
- (17)
- a surface of constant negative curvature immersed in ${H}_{s}^{m}(-1)\subset {\mathbb{E}}_{s+1}^{m+1}$ as$$\begin{array}{c}r\left(\phantom{\rule{-2.168pt}{0ex}}sinh\left(\frac{2u}{\sqrt{3}}\right)-\frac{{v}^{2}}{3}-\left(\frac{7}{8}+\frac{{v}^{4}}{18}\right){e}^{\frac{2u}{\sqrt{3}}},\phantom{\rule{0.166667em}{0ex}}v+\left(\frac{{v}^{3}}{3}-\frac{v}{4}\right){e}^{\frac{2u}{\sqrt{3}}},\frac{1}{2}+\phantom{\rule{4.pt}{0ex}}\frac{{v}^{2}}{2}{e}^{\frac{2u}{\sqrt{3}}},\right.\hfill \\ \left.\phantom{\rule{28.90755pt}{0ex}}0,\dots ,0,v+\left(\phantom{\rule{4.pt}{0ex}}\frac{{v}^{3}}{3}+\frac{v}{4}\right){e}^{\frac{2u}{\sqrt{3}}},\phantom{\rule{0.166667em}{0ex}}sinh\left(\frac{2u}{\sqrt{3}}\right)-\frac{{v}^{2}}{3}-\left(\frac{1}{8}+\frac{{v}^{4}}{18}\right){e}^{\frac{2u}{\sqrt{3}}},\frac{\sqrt{{r}^{2}-1}}{r}\right)\hfill \end{array}$$
- (18)
- a surface of constant negative curvature immersed in ${H}_{2}^{4}(-1)\subset {H}_{s}^{m}(-1)\subset {\mathbb{E}}_{s+1}^{m+1}$ defined as$$\begin{array}{c}r\left(\phantom{\rule{-2.168pt}{0ex}}sinh\left(\frac{2u}{\sqrt{3}}\right)-\frac{{v}^{2}}{3}-\left(\frac{7}{8}+\frac{{v}^{4}}{18}\right){e}^{\frac{2u}{\sqrt{3}}},\phantom{\rule{0.166667em}{0ex}}v+\left(\frac{{v}^{3}}{3}-\frac{v}{4}\right){e}^{\frac{2u}{\sqrt{3}}},\phantom{\rule{4.pt}{0ex}}\frac{1}{2}+\frac{{v}^{2}}{2}{e}^{\frac{2u}{\sqrt{3}}},\right.\hfill \\ \hfill \left.\phantom{\rule{28.90755pt}{0ex}}\frac{\sqrt{1-{r}^{2}}}{r},0,\dots ,0,v+\left(\frac{{v}^{3}}{3}+\frac{v}{4}\right){e}^{\frac{2u}{\sqrt{3}}},\phantom{\rule{0.166667em}{0ex}}sinh\left(\frac{2u}{\sqrt{3}}\right)-\frac{{v}^{2}}{3}-\left(\frac{1}{8}+\frac{{v}^{4}}{18}\right){e}^{\frac{2u}{\sqrt{3}}}\right)\end{array}$$

- (B)
- $L=({f}_{1},\dots ,{f}_{\ell},\varphi ,{f}_{\ell},\dots ,{f}_{1})$, where ${f}_{1},\dots ,{f}_{\ell}$ are polynomials of degree $\le 2$ in $u,v$ and ϕ is a surface given by (5), (7), (8) or (11)–(18) from $\left(A\right)$ or
- (C)
- $L=(r,\varphi ,r)$, where r is a positive number and ϕ is a surface given by (1)–(4), (6), (9) or (10) from $\left(A\right)$.

#### 12.2. A Parallel Spatial Surfaces in ${H}_{2}^{4}$

**Theorem**

**33.**

**Remark**

**5.**

#### 12.3. Special Case: Parallel Surfaces in ${H}_{1}^{3}$

**Corollary**

**3.**

- (i)
- a hyperbolic plane ${H}^{2}$ defined by $(a,bcoshucoshv,bcoshusinhv,bsinhu)$, ${a}^{2}+{b}^{2}=1$;
- (ii)
- a surface ${H}^{1}\times {H}^{1}$ defined by $(acoshu,bcoshv,asinhu,bsinhv)$, ${a}^{2}+{b}^{2}=1$.

**Remark**

**6.**

## 13. Parallel Lorentz Surfaces in Pseudo-Euclidean Spaces

**Theorem**

**34.**

#### 13.1. Classification of Parallel Lorentzian Surfaces in ${\mathbb{E}}_{s}^{m}$

**Theorem**

**35.**

- (A)
- the surface is an open portion of one of the following fifteen types of surfaces:
- (1)
- a totally geodesic plane ${\mathbb{E}}_{1}^{2}\subset {\mathbb{E}}_{s}^{m}$ given by $(x,y)\in {\mathbb{E}}_{1}^{2}\subset {\mathbb{E}}_{s}^{m}$;
- (2)
- a totally umbilical de Sitter space ${S}_{1}^{2}$ in a totally geodesic ${\mathbb{E}}_{1}^{3}\subset {\mathbb{E}}_{s}^{m}$ given by$$(sinhx,coshxcosy,coshxsiny);$$
- (3)
- a flat cylinder ${\mathbb{E}}_{1}^{1}\times {S}^{1}$ in a totally geodesic ${\mathbb{E}}_{1}^{3}\subset {\mathbb{E}}_{s}^{m}$ given by $\left(x,cosy,siny\right)$;
- (4)
- a flat cylinder ${S}_{1}^{1}\times {\mathbb{E}}^{1}$ in a totally geodesic ${\mathbb{E}}_{1}^{3}\subset {\mathbb{E}}_{s}^{m}$ given by $\left(sinhx,coshx,y\right)$;
- (5)
- a flat minimal surface in a totally geodesic ${\mathbb{E}}_{1}^{3}\subset {\mathbb{E}}_{s}^{m}$ given by$$\left(\frac{1}{6}{(x-y)}^{3}+x,\frac{1}{6}{(x-y)}^{3}+y,\frac{1}{2}{(x-y)}^{2}\right);$$
- (6)
- a flat surface ${S}_{1}^{1}\times {S}^{1}$ in a totally geodesic ${\mathbb{E}}_{1}^{4}\subset {\mathbb{E}}_{s}^{m}$ given by $\left(asinhx,acoshx,bcosy,bsiny\right),$ with $a,b>0;$
- (7)
- an anti-de Sitter space ${H}_{1}^{2}$ in a totally geodesic ${\mathbb{E}}_{2}^{3}\subseteq {\mathbb{E}}_{s}^{m}$ given by $(sinx,cosxcoshy,cosxsinhy);$
- (8)
- a flat minimal surface in a totally geodesic ${\mathbb{E}}_{2}^{3}\subseteq {\mathbb{E}}_{s}^{m}$ defined by$$\left(\frac{{a}^{2}{x}^{2}}{2},\frac{x}{2}-\frac{{a}^{4}{x}^{2}}{6}+y,\frac{x}{2}+\frac{{a}^{4}{x}^{2}}{6}-y\right),\phantom{\rule{0.277778em}{0ex}}a>0;$$
- (9)
- a non-minimal flat surface in a totally geodesic ${\mathbb{E}}_{2}^{3}\subseteq {\mathbb{E}}_{s}^{m}$ defined by$$\begin{array}{c}(\frac{1}{2b}cos\left(\frac{\sqrt{2b}}{a}({a}^{2}x+by)\right),\frac{1}{2b}sin\left(\frac{\sqrt{2b}}{a}({a}^{2}x+by)\right),\frac{{a}^{2}x-by}{a\sqrt{2b}}),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}a,b>0;\hfill \end{array}$$
- (10)
- a non-minimal flat surface in a totally geodesic ${\mathbb{E}}_{2}^{3}\subseteq {\mathbb{E}}_{s}^{m}$ defined by$$\begin{array}{c}(\frac{{a}^{2}x+by}{a\sqrt{2b}},\frac{1}{2b}cosh\left(\phantom{\rule{-0.166667em}{0ex}}\frac{\sqrt{2b}}{a}({a}^{2}x-by)\right),\frac{1}{2b}sinh\left(\frac{\sqrt{2b}}{a}({a}^{2}x-by)\right)),\phantom{\rule{0.277778em}{0ex}}a,b>0;\hfill \end{array}$$
- (11)
- a flat surface ${H}_{1}^{1}\times {H}^{1}$ in a totally geodesic ${\mathbb{E}}_{2}^{4}\subset {\mathbb{E}}_{s}^{m}$ given by $\left(asinhx,bcoshv,acoshx,bsinhy\right)$ with $a,b>0;$
- (12)
- a marginally trapped flat surface in a totally geodesic ${\mathbb{E}}_{2}^{4}\subseteq {\mathbb{E}}_{s}^{m}$ defined by$$\begin{array}{c}(acosxcoshy+bsinxsinhy,asinxcoshy-bcosxsinhy,\hfill \\ bcosxcoshy-asinxsinhy,bsinxcoshy+acosxsinhy),\phantom{\rule{0.277778em}{0ex}}a,b\in \mathbf{R};\hfill \end{array}$$
- (13)
- a marginally trapped flat surface in a totally geodesic ${\mathbb{E}}_{2}^{4}\subseteq {\mathbb{E}}_{s}^{m}$ given by$$\begin{array}{c}\left(\right(1+a)siny-(x+ay)cosy,(1+a)cosy+(x+ay)siny,\hfill \\ (1-a)siny+(x+ay)cosy,(1-a)cosy-(x+ay)siny),\phantom{\rule{0.277778em}{0ex}}a\in \mathbf{R};\hfill \end{array}$$
- (14)
- a non-minimal flat surface in a totally geodesic ${\mathbb{E}}_{3}^{4}\subseteq {\mathbb{E}}_{s}^{m}$ defined by$$\begin{array}{c}\left(cos\left(\frac{\sqrt{b}({a}^{3}x+by)}{{a}^{5/2}}\right)\phantom{\rule{-0.166667em}{0ex}},sin\left(\frac{\sqrt{b}({a}^{3}x+by)}{{a}^{5/2}}\right)\phantom{\rule{-0.166667em}{0ex}},cosh\left(\frac{\sqrt{b}({a}^{3}x-by)}{{a}^{5/2}}\right)\phantom{\rule{-0.166667em}{0ex}},sinh\left(\frac{\sqrt{b}({a}^{3}x-by)}{{a}^{5/2}}\right)\right),\hfill \end{array}$$
- (15)
- a non-minimal flat surface in a totally geodesic ${\mathbb{E}}_{3}^{4}\subseteq {\mathbb{E}}_{s}^{m}$ defined by$$\begin{array}{c}(\frac{\sqrt[4]{{\delta}^{2}+{\phi}^{2}}cos\left(\lambda \left(bx+\sqrt{{\delta}^{2}+{\phi}^{2}}y\right)\right)}{\sqrt{2}b\sqrt{\sqrt{{\delta}^{2}+{\phi}^{2}}+\delta}},\frac{\sqrt[4]{{\delta}^{2}+{\phi}^{2}}sin\left(\lambda \left(bx+\sqrt{{\delta}^{2}+{\phi}^{2}}y\right)\right)}{\sqrt{2}b\sqrt{\sqrt{{\delta}^{2}+{\phi}^{2}}+\delta}},\hfill \\ \frac{\sqrt[4]{{\delta}^{2}+{\phi}^{2}}cosh\left(\mu \left(bx-sqrt{\delta}^{2}+{\phi}^{2}y\right)\right)}{\sqrt{2}b\sqrt{\sqrt{{\delta}^{2}+{\phi}^{2}}-\delta}},\frac{\phantom{\rule{-0.166667em}{0ex}}\sqrt[4]{{\delta}^{2}+{\phi}^{2}}sin\left(\mu \left(bx-\sqrt{{\delta}^{2}+{\phi}^{2}}y\right)\right)}{\sqrt{2}b\sqrt{\phantom{\rule{-0.166667em}{0ex}}\sqrt{{\delta}^{2}+{\phi}^{2}}-\delta}})\hfill \end{array}$$$$\begin{array}{c}\lambda =\frac{\sqrt{b\sqrt{{\delta}^{2}+{\phi}^{2}}+b\delta}}{\sqrt{{\delta}^{2}+{\phi}^{2}}},\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mu =\frac{\sqrt{b\sqrt{{\delta}^{2}+{\phi}^{2}}-b\delta}}{\sqrt{{\delta}^{2}+{\phi}^{2}}},\hfill \end{array}$$

or - (B)
- ${M}_{1}^{2}$ is a flat surface and the immersion takes the form: $({f}_{1},\dots ,{f}_{\ell},\varphi (x,y),{f}_{\ell},\dots ,{f}_{1}),$ where $\varphi =\varphi (x,y)$ is given by one of (1), (3)–(6), (8)–(15) and ${f}_{1},\dots ,{f}_{\ell}\phantom{\rule{0.166667em}{0ex}}(\ell \ge 1)$ are polynomials of degree $\le 2$ in $x,y$.

#### 13.2. Classification of Parallel Lorentzian Surfaces in ${\mathbb{E}}_{1}^{3}$

**Corollary**

**4.**

- (1)
- the Lorentzian plane ${E}_{1}^{2}:L(u,v)=(u,v,0)$;
- (2)
- a de Sitter space ${S}_{1}^{2}:L(u,v)=a(sinhu,coshucosv,coshusinv),a>0$;
- (3)
- a cylinder ${\mathbb{E}}_{1}^{1}\times {S}^{1}:L(u,v)=(u,acosv,asinv),\phantom{\rule{0.166667em}{0ex}}a>0$;
- (4)
- a cylinder ${S}_{1}^{1}\times {\mathbb{E}}^{1}:L(u,v)=(asinhu,acoshu,v),\phantom{\rule{0.166667em}{0ex}}a>0$;
- (5)
- the null scroll ${\mathbb{N}}_{1}^{2}$ with rulings in the direction of $(1,1,0)$ of the null cubic given by $\alpha \left(u\right)=\left(\frac{4}{3}{u}^{3}+u,\frac{4}{3}{u}^{3}-u,2{u}^{2}\right)$.

**Remark**

**7.**

## 14. Parallel Surfaces in a Light Cone $\mathcal{L}\mathcal{C}$

#### 14.1. Light Cones in General Relativity

#### 14.2. Parallel Surfaces in ${\mathcal{L}C}_{1}^{3}\subset {\mathbb{E}}_{1}^{4}$

**Theorem**

**36.**

- (1)
- a totally umbilical surface of positive curvature given by $a(1,cosucosv,cosusinv,sinu),\phantom{\rule{0.277778em}{0ex}}a>0;$
- (2)
- totally umbilical surface of negative curvature given by $a(coshucoshv,coshusinhv,sinhu,1),\phantom{\rule{0.277778em}{0ex}}a>0;$
- (3)
- a flat totally umbilical surface given by $\left({u}^{2}+{v}^{2}+\frac{1}{4},{u}^{2}+{v}^{2}-\frac{1}{4},u,v\right);$
- (4)
- a flat surface given by $a(coshu,sinhu,cosv,sinv),a>0.$

#### 14.3. Parallel Surfaces in ${\mathcal{L}C}_{2}^{3}\subset {\mathbb{E}}_{2}^{4}$

**Theorem**

**37.**

- (1)
- a totally umbilical surface of positive curvature given by $a(sinhu,1,coshucosv,coshusinv),\phantom{\rule{0.277778em}{0ex}}a>0;$
- (2)
- a totally umbilical surface of negative curvature given by $a(sinu,cosucoshv,1,cosusinhv),a>0;$
- (3)
- a totally umbilical flat surface defined by$$\left(u,{u}^{2}+{v}^{2}-\frac{1}{4},{u}^{2}+{v}^{2}+\frac{1}{4},v\right);$$
- (4)
- a flat surface defined by $a(sinhu,coshv,coshu,sinhv),a>0;$
- (5)
- a flat surface defined by $a(sinu,cosu,cosv,sinv),a>0;$
- (6)
- a flat surface defined by$$\begin{array}{c}a(sinhucosv+sinhusinv,coshusinv-sinhucosv,\hfill \end{array}$$$$\begin{array}{c}\hfill \phantom{\rule{14.45377pt}{0ex}}coshucosv-sinhusinv,coshusinv+sinhucosv),\phantom{\rule{0.277778em}{0ex}}a>0;\end{array}$$
- (7)
- a flat surface defined by $a(cosv-usinv,sinv+ucosv,cosv+usinv,sinv-ucosv),a>0;$
- (8)
- a flat surface defined by $a(coshu-vsinhu,sinhu+vcoshu,coshu+vsinhu,sinhu-vcoshu)$ with $a>0.$

## 15. Parallel Surfaces in De Sitter Space-Time ${S}_{1}^{4}$

#### 15.1. Classification of Parallel Spatial Surfaces in De Sitter Space-Time ${S}_{1}^{4}$

**Theorem**

**38.**

- (1)
- a totally umbilical sphere ${S}^{2}$ given locally by $(c,bcosucosv,bcosusinv,bsinu,a),\phantom{\rule{0.277778em}{0ex}}{a}^{2}+{b}^{2}-{c}^{2}=1;$
- (2)
- a totally umbilical hyperbolic plane ${H}^{2}$ given by $(acoshucoshv,acoshusinhv,asinhu,b,c)$ with ${b}^{2}+{c}^{2}-{a}^{2}=1;$
- (3)
- a torus ${S}^{1}\times {S}^{1}$ given by $(a,bcosu,bsinu,ccosv,csinv)$ with ${b}^{2}+{c}^{2}-{a}^{2}=1;$
- (4)
- a flat surface ${H}^{1}\times {S}^{1}$ given by $(bcoshu,bsinhu,ccosv,csinv,a)$ with ${a}^{2}+{c}^{2}-{b}^{2}=1;$
- (5)
- a totally umbilical flat surface defined by$$\left({u}^{2}+{v}^{2}+{a}^{2}+\frac{1}{4},{u}^{2}+{v}^{2}+{a}^{2}-\frac{1}{4},u,v,\sqrt{1+{a}^{2}}\right);$$
- (6)
- a flat surface defined by$$\left({v}^{2}-\frac{3}{4}+{a}^{2},acosu,asinu,v,{v}^{2}-\frac{5}{4}+{a}^{2}\right),\phantom{\rule{0.277778em}{0ex}}a>0;$$
- (7)
- a flat surface defined by$$\frac{1}{\sqrt{1+{a}^{2}}}\left({u}^{2}+{v}^{2}-\frac{3}{4},{u}^{2}+{v}^{2}-\frac{5}{4},u,v,a\right),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}a\in \mathbf{R};$$
- (8)
- a marginally trapped flat surface defined by $\frac{1}{2}\left(2{u}^{2}-1,2{u}^{2}-2,2u,sinv,cosv\right);$
- (9)
- a marginally trapped flat surface defined by$$\left(\frac{b}{\sqrt{4-{b}^{2}}},\frac{cosu}{\sqrt{2-b}},\frac{sinu}{\sqrt{2-b}},\frac{cosv}{\sqrt{2+b}},\frac{sinv}{\sqrt{2+b}}\right);\phantom{\rule{0.277778em}{0ex}}\left|b\right|<2;$$
- (10)
- a marginally trapped flat surface defined by$$\left(\frac{coshu}{\sqrt{b-2}},\frac{sinhu}{\sqrt{b-2}},\frac{cosv}{\sqrt{2+b}},\frac{sinv}{\sqrt{2+b}},\frac{b}{\sqrt{{b}^{2}-4}}\right);\phantom{\rule{0.277778em}{0ex}}b>2.$$

**Corollary**

**5.**

- (1)
- a totally umbilical sphere ${S}^{2}$ given locally by $(a,bsinu,bcosucosv,bcosusinv)$ with ${b}^{2}-{a}^{2}={c}^{-1};$
- (2)
- a totally umbilical Euclidean ${\mathbb{E}}^{2}$ plane given by $\frac{1}{\sqrt{c}}\left({u}^{2}+{v}^{2}-\frac{3}{4},{u}^{2}+{v}^{2}-\frac{5}{4},u,v\right);$
- (3)
- a totally umbilical hyperbolic plane ${H}^{2}$ given by $(acoshucoshv,acoshusinhv,asinhu,b),$ with ${b}^{2}-{a}^{2}={c}^{-1};$
- (4)
- a flat surface ${H}^{1}\times {S}^{1}$ given by $(acoshu,asinhu,bcosv,bsinv)$ with ${a}^{2}+{b}^{2}={c}^{-1}.$

#### 15.2. Classification of Parallel Lorentzian Surfaces in De Sitter Space-Time ${S}_{1}^{4}$

**Theorem**

**39.**

- (1)
- a totally umbilical de Sitter space ${S}_{1}^{2}$ in ${S}_{1}^{4}\left(1\right)$ given by $(asinhu,acoshucosv,acoshusinv,b,0)$ with ${a}^{2}+{b}^{2}=1;$
- (2)
- a flat surface ${S}_{1}^{1}\times {S}^{1}$ given by $(asinhu,acoshu,bcosv,bsinv,0),{a}^{2}+{b}^{2}=1.$

## 16. Parallel Surfaces in Anti-De Sitter Space-Time ${H}_{1}^{4}$

#### 16.1. Classification of Parallel Spatial Surfaces in ${H}_{1}^{4}$

**Theorem**

**40.**

- (1)
- a totally umbilical sphere ${S}^{2}$ given locally by $(a,c,bsinu,bcosucosv,bcosusinv)$, ${a}^{2}-{b}^{2}+{c}^{2}=1;$
- (2)
- a totally umbilical hyperbolic plane ${H}^{2}$ given locally by $(a,bcoshucoshv,bcoshusinhv,bsinhu,c)$ with ${a}^{2}+{b}^{2}-{c}^{2}=1;$
- (3)
- flat surface ${H}^{1}\times {S}^{1}$ given by $(a,bcoshu,bsinhu,ccosv,csinv)$ with ${a}^{2}+{b}^{2}-{c}^{2}=1;$
- (4)
- a flat surface ${H}^{1}\times {H}^{1}$ given by $(bcoshu,ccoshv,bsinhu,csinhv,a)$ with ${b}^{2}+{c}^{2}-{a}^{2}=1;$
- (5)
- a totally umbilical flat surface defined by$$\left(\sqrt{1-{a}^{2}},{u}^{2}+{v}^{2}+{a}^{2}+\frac{1}{4},{u}^{2}+{v}^{2}+{a}^{2}-\frac{1}{4},u,v\right),\phantom{\rule{0.277778em}{0ex}}a\in (0,1);$$
- (6)
- a flat surface defined by$$\left(a,b\left({u}^{2}+{v}^{2}-\frac{3}{4}\right),b\left({u}^{2}+{v}^{2}-\frac{5}{4}\right),bu,bv\right),\phantom{\rule{0.277778em}{0ex}}{a}^{2}=1+{b}^{2}>1;$$
- (7)
- a flat surface defined by$$\left({v}^{2}+\frac{5}{4}-{a}^{2},acoshu,asinhu,v,{v}^{2}+\frac{3}{4}-{a}^{2}\right),\phantom{\rule{0.277778em}{0ex}}a\ne 0;$$
- (8)
- the marginally trapped flat surface defined by$$\left({u}^{2}+1,\frac{1}{2}coshv,u,\frac{1}{2}sinhv,{u}^{2}+\frac{1}{2}\right);$$
- (9)
- a marginally trapped flat surface defined by$$\left(\frac{coshu}{\sqrt{2-b}},\frac{coshv}{\sqrt{2+b}},\frac{sinhu}{\sqrt{2-b}},\frac{sinhv}{\sqrt{2+b}},\frac{b}{\sqrt{4-{b}^{2}}}\right),\phantom{\rule{0.277778em}{0ex}}\left|b\right|<2;$$
- (10)
- a flat marginally trapped surface defined by$$\left(\frac{b}{\sqrt{{b}^{2}-4}},\frac{coshv}{\sqrt{b+2}},\frac{sinhu}{\sqrt{b+2}},\frac{cosu}{\sqrt{b-1b}},\frac{sinu}{\sqrt{b-2}}\right),\phantom{\rule{0.277778em}{0ex}}b>2.$$

**Corollary**

**6.**

- (1)
- a hyperbolic plane ${H}^{2}$ defined by $(a,bcoshucoshv,bcoshusinhv,bsinhu),\phantom{\rule{0.277778em}{0ex}}{a}^{2}+{b}^{2}=1;$
- (2)
- a surface ${H}^{1}\times {H}^{1}$ defined by $(acoshu,bcoshv,asinhu,bsinhv),{a}^{2}+{b}^{2}=1.$

#### 16.2. Classification of Parallel Lorentzian Surfaces in Anti-De Sitter Space-Time ${H}_{1}^{4}$

**Theorem**

**41.**

- (1)
- a totally umbilical de Sitter space ${S}_{1}^{2}$ given by $(c,asinhucosv,acoshucosv,acoshusinb,b)$ with ${c}^{2}-{a}^{2}-{b}^{2}=1;$
- (2)
- a totally umbilical anti-de Sitter space ${H}_{1}^{2}$ given by $(asinu,acosucoshv,acosusinhv,0,b)$ with ${a}^{2}-{b}^{2}=1;$
- (3)
- a flat surface ${S}_{1}^{1}\times {H}^{1}$ given by $(c,asinhu,acoshucosv,acoshusinv,b)$ with ${c}^{2}-{a}^{2}-{b}^{2}=1;$
- (4)
- a flat surface ${H}_{1}^{1}\times {S}^{1}$ given by $(acosu,asinu,bcosv,bsinv,c)$ with ${a}^{2}+{b}^{2}-{c}^{2}=1;$
- (5)
- a flat surface ${S}_{1}^{1}\times {S}^{1}$ given by $(a,bsinhu,bcoshu,ccosv,csinv)$ with ${a}^{2}-{b}^{2}-{c}^{2}=1;$
- (6)
- a totally umbilical flat surface defined by $\left({u}^{2}-{v}^{2}-\frac{5}{4},au,av,a\left({u}^{2}-{v}^{2}-\frac{3}{4}\right),b\right)$ with ${a}^{2}-{b}^{2}=1;$
- (7)
- a flat surface defined by$$\left(acosv-\frac{a(u-v)}{2}sinv,asinv+\frac{a(u-v)}{2}cosv,\frac{a(u-v)}{2}sinv,\frac{a(u-v)}{2}cosv,b\right),\phantom{\rule{0.277778em}{0ex}}a\in \mathbf{R};$$
- (8)
- a flat surface defined by$$\left(acoshv-\frac{a(u+v)}{2}sinhv,\frac{a(u+v)}{2}coshv,asinhv-\frac{a(u+v)}{2}coshv,\frac{a(u+v)}{2}sinhv,b\right)$$
- (9)
- a surface defined by$$\begin{array}{c}(acosucoshv-atanksinusinhv,asecksinucoshv,\hfill \\ \phantom{\rule{14.45377pt}{0ex}}acosusinhv-atanksinucoshv,asecksinusinhv,b),\hfill \end{array}$$
- (10)
- a surface defined by$$\left(\frac{{b}^{2}({u}^{2}-{k}^{2}-1)-1}{2{b}^{2}k},u,\frac{cosbv}{b},\frac{sinbv}{b},\frac{{b}^{2}({u}^{2}+{k}^{2}-1)-1}{2{b}^{2}k}\right),\phantom{\rule{0.277778em}{0ex}}b,k\ne 1;$$
- (11)
- a surface defined by$$\left(\frac{-{a}^{2}({v}^{2}+{k}^{2}+1)+1}{2{a}^{2}k},\frac{sinhau}{a},\frac{coshau}{a},v,\frac{{a}^{2}({k}^{2}-{v}^{2}-1)-1}{2{a}^{2}k}\right),\phantom{\rule{0.277778em}{0ex}}a,k\ne 1;$$
- (12)
- a surface defined by$$\begin{array}{c}(\frac{{(u-v)}^{4}}{24k}+\frac{{u}^{2}-{v}^{2}-{k}^{2}-1}{2k},\frac{1}{6}{(u-v)}^{3}+u,\frac{1}{2}{(u-v)}^{2},\hfill \end{array}$$$$\begin{array}{c}\hfill \phantom{\rule{14.45377pt}{0ex}}\frac{1}{6}{(u-v)}^{3}+v,\frac{{(u-v)}^{4}}{24k}+\frac{{u}^{2}-{v}^{2}+{k}^{2}-1}{2k}),\phantom{\rule{0.277778em}{0ex}}k\ne 0.\end{array}$$

#### 16.3. Special Case: Parallel Lorentzian Surfaces in ${H}_{1}^{3}$

**Corollary**

**7.**

- (1)
- a de Sitter space ${S}_{1}^{2}$ defined by $(a,bsinhu,bcoshusinv,bcoshucosv)$ with ${a}^{2}-{b}^{2}=1;$
- (2)
- the surface $\left({u}^{2}-{v}^{2}-\frac{5}{4},u,v,{u}^{2}-{v}^{2}-\frac{3}{4}\right);$
- (3)
- an anti-de Sitter space ${H}_{1}^{2}$ defined by $(asinu,acosucoshv,acosusinhv,b)$ with ${a}^{2}-{b}^{2}=1;$
- (4)
- a surface ${S}_{1}^{1}\times {H}^{1}$ defined by $(asinhu,bcoshv,acoshu,bsinhv)$ with ${b}^{2}-{a}^{2}=1;$
- (5)
- a surface ${H}_{1}^{1}\times {S}^{1}$ defined by $(acosu,asinu,bcosv,bsinv)$ with ${a}^{2}-{b}^{2}=1;$
- (6)
- a surface defined by$$\begin{array}{c}(cosucoshv-tanksinusinhv,secksinucoshv,\hfill \\ cosusinhv-tanksinucoshv,secksinusinhv),cosk\ne 0\hfill \end{array}$$
- (7)
- the surface defined by$$\left(cosv-\frac{u-v}{2}sinv,sinv+\frac{u-v}{2}cosv,\frac{u-v}{2}sinv,\frac{u-v}{2}cosv\right);$$
- (8)
- the surface defined by$$\left(coshv-\frac{u+v}{2}sinhv,\frac{u+v}{2}coshv,sinhv-\frac{u+v}{2}coshv,\frac{u+v}{2}sinhv\right).$$

## 17. Parallel Spatial Surfaces in ${S}_{2}^{4}$

#### 17.1. Four-Dimensional Manifolds with Neutral Metrics

#### 17.2. Classification of Parallel Lorentzian Surfaces in ${S}_{2}^{4}$

**Theorem**

**42.**

- (1)
- a totally geodesic de Sitter space-time ${S}_{1}^{2}\left(1\right)\subset {S}_{2}^{4}\left(1\right)\subset {\mathbb{E}}_{2}^{5}$;
- (2)
- a flat surface in a totally geodesic ${S}_{1}^{3}\left(1\right)\subset {S}_{2}^{4}\left(1\right)$ defined by$$\begin{array}{c}\left(\sqrt{{a}^{2}+{b}^{2}-1},asinhu,acoshu,bcosv,bsinv\right),\phantom{\rule{0.277778em}{0ex}}a,b>0,{a}^{2}+{b}^{2}\ge 1;\hfill \end{array}$$
- (3)
- a flat surface defined by$$\begin{array}{c}\phantom{\rule{2.8903pt}{0ex}}(acosusinhv+bsinucoshv,\sqrt{{a}^{2}+{b}^{2}}sinusinhv,\sqrt{{a}^{2}+{b}^{2}}sinucoshv,\hfill \\ \phantom{\rule{21.68121pt}{0ex}}acosucoshv+bsinusinhv,\sqrt{1-{a}^{2}}\phantom{\rule{0.166667em}{0ex}}),\phantom{\rule{0.277778em}{0ex}}a\in (0,1];\hfill \end{array}$$
- (4)
- a flat surface defined by $\left(acosu,asinu,bcosv,bsinv,\sqrt{1+{a}^{2}-{b}^{2}}\right),\phantom{\rule{0.277778em}{0ex}}a,b>0,\phantom{\rule{0.277778em}{0ex}}{b}^{2}\le 1+{a}^{2};$
- (5)
- a flat surface defined by$$\begin{array}{c}\left(ku,p{u}^{2}+\frac{(1-{b}^{2})\phi}{{k}^{2}}-\frac{{k}^{2}}{4\phi},bsinv,bcosv,p{u}^{2}+\frac{(1-{b}^{2})\phi}{{k}^{2}}+\frac{{k}^{2}}{4\phi}\right),\phantom{\rule{0.277778em}{0ex}}b,k,p,\phi \ne 0;\hfill \end{array}$$
- (6)
- a flat surface defined by $\left(\sqrt{{b}^{2}-{a}^{2}-1},acoshu,asinhu,bcosv,bsinv\right),\phantom{\rule{0.277778em}{0ex}}a,b>0,\phantom{\rule{0.277778em}{0ex}}{b}^{2}\ge 1+{a}^{2};$
- (7)
- a flat surface defined by$$\begin{array}{c}\left(p{u}^{2}+\frac{({b}^{2}-1)\phi}{{k}^{2}}+\frac{{k}^{2}}{4\phi},bsinhv,bcoshv,ku,p{u}^{2}+\frac{({b}^{2}-1)\phi}{{k}^{2}}-\frac{{k}^{2}}{4\phi}\right),\phantom{\rule{0.277778em}{0ex}}b,k,p,\phi \ne 0;\hfill \end{array}$$
- (8)
- a flat surface given by $\left(acoshu,bsinhv,asinhu,bcoshv,\sqrt{1+{a}^{2}-{b}^{2}}\right),\phantom{\rule{0.166667em}{0ex}}a,b>0,\phantom{\rule{0.166667em}{0ex}}{b}^{2}\le 1+{a}^{2};$
- (9)
- a marginally trapped surface of constant curvature one defined by$$\left(\frac{xy}{x+y},\frac{2}{x+y},\frac{x-y}{x+y},\frac{2+xy}{x+y},0\right),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}x+y\ne 0;$$
- (10)
- a flat surface defined by $\left(x+xy,y-xy,x-y+xy,1+xy,0\right);$
- (11)
- a surface of positive curvature ${c}^{2}$ defined by$$\left(\frac{xy-{c}^{2}}{{c}^{2}(x+y)},\frac{2\sqrt{1-{c}^{2}}\phantom{\rule{0.166667em}{0ex}}y}{{c}^{2}(x+y)},\frac{xy+{c}^{2}}{{c}^{2}(x+y)},\frac{{c}^{2}(x+y)-2y}{{c}^{2}(x+y)},0\right),\phantom{\rule{0.277778em}{0ex}}c\in (0,1),\phantom{\rule{0.277778em}{0ex}}x+y\ne 0;$$
- (12)
- a surface of positive curvature ${c}^{2}$ defined by$$\left(0,\frac{xy-{c}^{2}}{{c}^{2}(x+y)},\frac{xy+{c}^{2}}{{c}^{2}(x+y)},\frac{{c}^{2}(x+y)-2y}{{c}^{2}(x+y)},\frac{2\sqrt{{c}^{2}-1}\phantom{\rule{0.166667em}{0ex}}y}{{c}^{2}(x+y)}\right),\phantom{\rule{0.277778em}{0ex}}c>1,\phantom{\rule{0.277778em}{0ex}}x+y\ne 0;$$
- (13)
- a surface of negative curvature $-{c}^{2}$ defined by$$\begin{array}{c}\frac{1}{c}\left(coshu-sinhutanhv,sinhutanhv,sinhu-coshutanhv,\sqrt{1+{c}^{2}},0\right),\phantom{\rule{0.166667em}{0ex}}c>0;\hfill \end{array}$$
- (14)
- a flat surface defined by$$\begin{array}{c}(\frac{1+8{c}^{2}+2v}{4c}cosu+\frac{1+v}{2c}sinu,\frac{4{c}^{2}-1}{4c}cosu+\left(c+\frac{v}{2c}\right)sinu,\hfill \\ \phantom{\rule{7.22743pt}{0ex}}\left(\frac{1}{4c}+2c+\frac{v}{2c}\right)cosu+\frac{vsinu}{2c},\frac{4{c}^{2}+1}{4c}cosu+\frac{1+2{c}^{2}+v}{2c}sinu,0),\phantom{\rule{0.277778em}{0ex}}c>0;\hfill \end{array}$$
- (15)
- a flat surface defined by$$\begin{array}{c}\left({e}^{u}-\frac{(2c-v){e}^{-u}}{8c},\frac{v{e}^{u}}{4}-\frac{{e}^{-u}}{2c},{e}^{u}+\frac{(2c-v){e}^{-u}}{8c},\frac{v{e}^{u}}{4}+\frac{{e}^{-u}}{2c},0\right),\phantom{\rule{0.277778em}{0ex}}c>0;\hfill \end{array}$$
- (16)
- a flat surface defined by$$\begin{array}{c}\left(x+\frac{y}{2}+\frac{2{c}^{2}{y}^{3}}{3},xy+\frac{{c}^{2}{y}^{4}}{6},x-\frac{y}{2}+\frac{2{c}^{2}{y}^{3}}{3},c{y}^{2},1+xy+\frac{{c}^{2}{y}^{4}}{6}\right),\phantom{\rule{0.277778em}{0ex}}c>0;\hfill \end{array}$$
- (17)
- a flat surface defined by$$\begin{array}{c}\hfill \left(avsinhu+bcoshu,avcoshu,avcoshu+bsinhu,avsinhu,\sqrt{1+{b}^{2}}\right),\phantom{\rule{0.166667em}{0ex}}a,b\ne 0;\end{array}$$
- (18)
- a flat surface defined by $(asinu-bvcosu,acosu+bvcosu,bvcosu,bvsinu,\sqrt{1+{a}^{2}}),\phantom{\rule{0.277778em}{0ex}}a,b\ne 0;$
- (19)
- a flat surface defined by$$\begin{array}{c}\left(vcosu+\frac{sinu}{c},vsinu-\frac{cosu}{c},vcosu-\frac{sinu}{c},vsinu+\frac{cosu}{c},1\right),\phantom{\rule{0.166667em}{0ex}}c>0;\hfill \end{array}$$
- (20)
- a flat surface defined by$$\begin{array}{c}\hfill \left(cosucosv-\frac{sinusinv}{c},cosusinv+\frac{sinucosv}{c},\right.cosucosv+\frac{sinusinv}{c},\hfill \\ \hfill \left.\phantom{\rule{57.81621pt}{0ex}}cosusinv-\frac{sinucosv}{c},1\phantom{\rule{-0.72229pt}{0ex}}\right),\phantom{\rule{0.277778em}{0ex}}c>0;\hfill \end{array}$$
- (21)
- a flat surface defined by$$\begin{array}{c}\left({e}^{v}cosu+\frac{{e}^{-v}sinu}{c},{e}^{-v}cosu-\frac{{e}^{v}sinu}{c},{e}^{v}cosu-\frac{{e}^{-v}sinu}{c},{e}^{-v}cosu+\frac{{e}^{v}sinu}{c},1\right),\phantom{\rule{0.277778em}{0ex}}c>0;\hfill \end{array}$$
- (22)
- a flat surface defined by $\left({e}^{u}+a{e}^{-u}v,{e}^{u}v-a{e}^{-u},{e}^{u}-a{e}^{-u}v,{e}^{u}v+a{e}^{-u},1\right),\phantom{\rule{0.277778em}{0ex}}a\ne 0;$
- (23)
- a flat surface defined by $\left({e}^{u}-a{e}^{-u},{e}^{v}+a{e}^{-v},{e}^{u}+a{e}^{-u},{e}^{v}-a{e}^{-v},1\right),\phantom{\rule{0.277778em}{0ex}}a\ne 0;$
- (24)
- a flat surface defined by $(acoshucosv,acoshusinv,asinhu,cosv,asinhusinv,\sqrt{1+{a}^{2}}),\phantom{\rule{0.166667em}{0ex}}a>0.$

#### 17.3. Classification of Parallel Lorentzian Surfaces in ${H}_{2}^{4}$

## 18. Parallel Spatial Surfaces in ${S}_{3}^{4}$ and in ${H}_{3}^{4}$

#### 18.1. Classification of Parallel Spatial Surfaces in ${S}_{3}^{4}$

#### 18.2. Classification of Parallel Spatial Surfaces in ${H}_{3}^{4}$

**Theorem**

**43.**

- (1)
- A totally geodesic anti-de Sitter space ${H}_{1}^{2}(-1)\subset {H}_{3}^{4}(-1)$;
- (2)
- A flat minimal surface in a totally geodesic ${H}_{2}^{3}(-1)\subset {H}_{3}^{4}(-1)$ defined by$$\begin{array}{c}\frac{1}{\sqrt{2}}\left(\phantom{\rule{-0.166667em}{0ex}}sin\left(ax+\frac{y}{a}\right)\phantom{\rule{-1.4457pt}{0ex}},cos\left(ax+\frac{y}{a}\right)\phantom{\rule{-1.4457pt}{0ex}},cosh\left(ax-\frac{y}{a}\right)\phantom{\rule{-1.4457pt}{0ex}},sinh\left(ax-\frac{y}{a}\right)\phantom{\rule{-1.4457pt}{0ex}},0\right),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}a>0;\hfill \end{array}$$
- (3)
- A totally umbilical anti-de Sitter space ${H}_{1}^{2}(-{c}^{2})$ in a totally geodesic ${H}_{2}^{3}(-1)\subset {H}_{3}^{4}(-1)$ given by$$\begin{array}{c}\phantom{\rule{0.0pt}{0ex}}\frac{1}{c}(0,\sqrt{{c}^{2}-1},tanh\left(\frac{cx+cy}{\sqrt{2}}\right),sinh\left(\sqrt{2}cy\right)tanh\left(\frac{cx+cy}{\sqrt{2}}\right)-cosh\left(\sqrt{2}cy\right),\hfill \\ \phantom{\rule{21.68121pt}{0ex}}sinh\left(\sqrt{2}cy\right)-cosh\left(\sqrt{2}cy\right)tanh\left(\frac{cx+cy}{\sqrt{2}}\right)),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}c>1;\hfill \end{array}$$
- (4)
- A CMC flat surface in a totally geodesic ${H}_{2}^{3}(-1)$ given by$$\begin{array}{c}\phantom{\rule{-43.36243pt}{0ex}}(\frac{\sqrt{\sqrt{1+{b}^{2}}-b}}{\sqrt{2}\sqrt[4]{1+{b}^{2}}}cos\phantom{\rule{-0.166667em}{0ex}}\left(\frac{\phantom{\rule{-0.166667em}{0ex}}\sqrt{\sqrt{1+{b}^{2}}+b}({a}^{2}x+\sqrt{1+{b}^{2}}y)}{a}\right),\hfill \\ \phantom{\rule{-28.90755pt}{0ex}}\frac{\sqrt{\sqrt{1+{b}^{2}}-b}}{\sqrt{2}\sqrt[4]{1\phantom{\rule{-0.166667em}{0ex}}+\phantom{\rule{-0.166667em}{0ex}}{b}^{2}}}sin\left(\frac{\sqrt{\phantom{\rule{-0.166667em}{0ex}}\sqrt{1+{b}^{2}}+b}({a}^{2}x+\sqrt{1+{b}^{2}}y)}{a}\right),\hfill \\ \phantom{\rule{-14.45377pt}{0ex}}\frac{\sqrt{\sqrt{1+{b}^{2}}+b}}{\sqrt{2}\sqrt[4]{1+{b}^{2}}}cosh\phantom{\rule{-0.166667em}{0ex}}\left(\frac{\sqrt{\sqrt{1+{b}^{2}}-b}({a}^{2}x-\sqrt{1+{b}^{2}}y)}{a}\right),\hfill \\ \frac{\sqrt{\sqrt{1+{b}^{2}}+b}}{\sqrt{2}\sqrt[4]{1+{b}^{2}}}sin\left(\frac{\sqrt{\sqrt{1+{b}^{2}}-b}({a}^{2}x-\sqrt{1+{b}^{2}}y)}{a}\right)\phantom{\rule{-0.166667em}{0ex}}),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}a,b,c>0;\hfill \end{array}$$
- (5)
- A non-minimal flat surface given by$$\begin{array}{c}\phantom{\rule{-21.68121pt}{0ex}}\frac{1}{\sqrt{2(1+{b}^{2})}}\left(\sqrt{2}b,cos\left(kx+\frac{{}^{}}{}\right)\right)\hfill \end{array}$$