# 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 $\left(\right)open="\{"\; close="\}">({z}_{0},{z}_{1},{z}_{2})\in C{P}^{2}\left(4\right)\phantom{\rule{4pt}{0ex}}|\phantom{\rule{4pt}{0ex}}{z}_{0}^{2}+{z}_{1}^{2}+{z}_{2}^{2}=0$ 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$$\begin{array}{c}\phantom{\rule{36.135pt}{0ex}}L(x,y)=(\frac{a\phantom{\rule{0.166667em}{0ex}}{e}^{-ix/a}}{\sqrt{1+{a}^{2}}},\frac{{e}^{i(ax+by)}}{\sqrt{1+{a}^{2}+{b}^{2}}}sin\left(\right)open="("\; close=")">\sqrt{1+{a}^{2}+{b}^{2}}\phantom{\rule{0.166667em}{0ex}}y,\hfill \end{array}\hfill \phantom{\rule{21.68121pt}{0ex}}\frac{{e}^{i(ax+by)}}{\sqrt{1+{a}^{2}}}\left(\right)open="("\; close=")">cos\left(\right)open="("\; close=")">\sqrt{1+{a}^{2}+{b}^{2}}\phantom{\rule{0.166667em}{0ex}}y-\frac{ib}{\sqrt{1+{a}^{2}+{b}^{2}}}sin\left(\right)open="("\; close=")">\sqrt{1+{a}^{2}+{b}^{2}}\phantom{\rule{0.166667em}{0ex}}y\\ ),$$

**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(\right)open="("\; close=")">cosh\left(\right)open="("\; close=")">\sqrt{1-{a}^{2}-{b}^{2}}\phantom{\rule{0.166667em}{0ex}}y-\frac{ib\phantom{\rule{0.166667em}{0ex}}sinh\left(\right)open="("\; close=")">\sqrt{1-{a}^{2}-{b}^{2}}\phantom{\rule{0.166667em}{0ex}}y}{}\sqrt{1-{a}^{2}-{b}^{2}}$$$\frac{{e}^{i(ax+by)}}{\sqrt{1-{a}^{2}-{b}^{2}}}sinh\left(\right)open="("\; close=")">\sqrt{1-{a}^{2}-{b}^{2}}\phantom{\rule{0.166667em}{0ex}}y,\frac{a\phantom{\rule{0.166667em}{0ex}}{e}^{ix/a}}{\sqrt{1-{a}^{2}}}),\phantom{\rule{0.277778em}{0ex}}a,b\in \mathbf{R},\phantom{\rule{0.277778em}{0ex}}a\ne 0,\phantom{\rule{0.277778em}{0ex}}{a}^{2}+{b}^{2}1;$$
- (2)
- $L(x,y)=\left(\right)open="("\; close=")">\phantom{\rule{-0.166667em}{0ex}}\left(\right)open="("\; close=")">\frac{i}{b}+y{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}}}$$\phantom{\rule{0.277778em}{0ex}}b\in \mathbf{R},\phantom{\rule{4pt}{0ex}}0<{b}^{2}<1;$
- (3)
- $L(x,y)=(\frac{{e}^{i(ax+by)}}{\sqrt{1-{a}^{2}}}\left(\right)open="("\; close=")">cos\left(\right)open="("\; close=")">\sqrt{{a}^{2}+{b}^{2}-1}\phantom{\rule{0.166667em}{0ex}}y-\frac{ib\phantom{\rule{0.166667em}{0ex}}sin\left(\right)open="("\; close=")">\sqrt{{a}^{2}+{b}^{2}-1}\phantom{\rule{0.166667em}{0ex}}y}{}\sqrt{{a}^{2}+{b}^{2}-1}$$$\frac{{e}^{i(ax+by)}}{\sqrt{{a}^{2}+{b}^{2}-1}}sin\left(\right)open="("\; close=")">\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}}},\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;$$
- (4)
- $L(x,y)=\left(\right)open="("\; close>\frac{a\phantom{\rule{0.166667em}{0ex}}{e}^{ix/a}}{\sqrt{{a}^{2}-1}},\frac{{e}^{i(ax+by)}}{\sqrt{{a}^{2}+{b}^{2}-1}}sin\left(\right)open="("\; close=")">\sqrt{{a}^{2}+{b}^{2}-1}\phantom{\rule{0.166667em}{0ex}}y,$$$\left(\right)open\; close=")">\frac{{e}^{i(ax+by)}}{\sqrt{{a}^{2}-1}}\left(\right)open="("\; close=")">cos\left(\right)open="("\; close=")">\sqrt{{a}^{2}+{b}^{2}-1}\phantom{\rule{0.166667em}{0ex}}y-\frac{ib\phantom{\rule{0.166667em}{0ex}}sin\left(\right)open="("\; close=")">\sqrt{{a}^{2}+{b}^{2}-1}\phantom{\rule{0.166667em}{0ex}}y}{}\sqrt{{a}^{2}+{b}^{2}-1}$$
- (5)
- $L(x,y)=\frac{{e}^{ix}}{8{b}^{2}}\left(\right)open="("\; close=")">i+8{b}^{2}(i+x)-4by,i+8{b}^{2}x-4by,4b{e}^{2iby}$, $\phantom{\rule{0.277778em}{0ex}}\mathbf{R}\ni b\ne 0;$
- (6)
- $L(x,y)={e}^{ix}\left(\right)open="("\; close=")">1+\frac{{y}^{2}}{2}-ix,y,\frac{{y}^{2}}{2}-ix.$

#### 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 $\left(\right)open="("\; close=")">0,\dots ,0,cosu,sinucosv,sinusinv$
- (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 $\left(\right)open="("\; close=")">0,\dots ,0,u,cosv,sinv$
- (iv)
- a flat torus ${S}^{1}\times {S}^{1}$ in a totally geodesic ${\mathbb{E}}^{4}$ given by $\left(\right)$ 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$$\begin{array}{c}\phantom{\rule{14.45377pt}{0ex}}\left(\right)open="("\; close=")">0,\dots ,0,\frac{vw}{\sqrt{3}},\frac{uw}{\sqrt{3}},\frac{uv}{\sqrt{3}},\frac{{u}^{2}-{v}^{2}}{2\sqrt{3}},\frac{1}{6}\left(\right)open="("\; close=")">{u}^{2}+{v}^{2}-2{w}^{2}\phantom{\rule{-1.4457pt}{0ex}}\hfill & ,\phantom{\rule{0.277778em}{0ex}}{u}^{2}+{v}^{2}+{w}^{2}=3;\end{array}$$
- (vi)
- a hyperbolic 2-plane ${H}^{2}$ in a totally geodesic ${\mathbb{E}}_{1}^{3}$ as $\left(\right)open="("\; close=")">coshu,0,\dots ,0,sinhucosv,sinhusinv$
- (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 $\left(\right)open="("\; close=")">coshu,0,\dots ,0,sinhu,v$
- (viii)
- a flat surface ${H}^{1}\times {S}^{1}$ in a totally geodesic ${\mathbb{E}}_{1}^{4}\subset {\mathbb{E}}_{s}^{m}$ given by$$\left(\right)$$
- (ix)
- a flat totally umbilical surface of a totally geodesic ${\mathbb{E}}_{1}^{4}\subset {\mathbb{E}}_{s}^{m}$ defined by$$\left(\right)open="("\; close=")">{u}^{2}+{v}^{2}+\frac{1}{4},0,\dots ,0,u,v,{u}^{2}+{v}^{2}-\frac{1}{4}$$
- (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$$\left(\right)open="("\; close=")">acoshu,bcoshv,0,\dots ,0,asinhu,bsinhv$$
- (xi)
- a surface of curvature $-\frac{1}{3}$ lying in a totally geodesic ${\mathbb{E}}_{3}^{5}\subset {\mathbb{E}}_{s}^{m}$ given by$$\begin{array}{c}(sinh\left(\frac{2s}{\sqrt{3}}\right)-\frac{{t}^{2}}{3}-\left(\right)open="("\; close=")">\frac{7}{8}+\frac{{t}^{4}}{18}{e}^{\frac{2s}{\sqrt{3}}},\phantom{\rule{0.166667em}{0ex}}t+\left(\right)open="("\; close=")">\frac{{t}^{3}}{3}-\frac{t}{4}\hfill & {e}^{\frac{2s}{\sqrt{3}}},\frac{1}{2}+\frac{{t}^{2}}{2}{e}^{\frac{2s}{\sqrt{3}}},\end{array}\phantom{\rule{25.29494pt}{0ex}}0,\dots ,0,t+\left(\right)open="("\; close=")">\frac{{t}^{3}}{3}+\frac{t}{4}{e}^{\frac{2s}{\sqrt{3}}},\phantom{\rule{0.166667em}{0ex}}sinh\left(\frac{2s}{\sqrt{3}}\right)-\frac{{t}^{2}}{3}-\left(\right)open="("\; close=")">\frac{1}{8}+\frac{{t}^{4}}{18}\hfill & {e}^{\frac{2s}{\sqrt{3}}}),\phantom{\rule{0.277778em}{0ex}}or$$

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

- (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$$\frac{1}{\sqrt{c}}\left(\right)open="("\; close=")">{u}^{2}+{v}^{2}-\frac{3}{4},{u}^{2}+{v}^{2}-\frac{5}{4},u,v$$

**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 $\left(\right)$ with $b>0;$
- (2)
- a totally umbilical ${H}^{2}$ immersed in ${H}_{s}^{m}(-1)\subset {\mathbb{E}}_{s+1}^{m+1}$ as$$\left(\right)open="("\; close=")">rcoshu,0,\dots ,0,rsinhucosv,rsinhusinv,\sqrt{{r}^{2}-1}\phantom{\rule{0.166667em}{0ex}}$$
- (3)
- a totally umbilical ${H}^{2}$ immersed in ${H}_{s}^{m}(-1)\subset {\mathbb{E}}_{s+1}^{m+1}$ as$$\phantom{\rule{2.em}{0ex}}\phantom{\rule{1.em}{0ex}}\left(\right)open="("\; close=")">rcoshu,\sqrt{1-{r}^{2}},0,\dots ,0,rsinhucosv,rsinhusinv\phantom{\rule{0.166667em}{0ex}}$$
- (4)
- a totally umbilical ${S}^{2}$ immersed in ${H}_{s}^{m}(-1)\subset {\mathbb{E}}_{s+1}^{m+1}$ as$$\left(\right)open="("\; close=")">\sqrt{1+{r}^{2}},0,\dots ,0,rsinu,rcosucosv,rcosusinv$$
- (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(\right)open="("\; close=")">\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}}\hfill \end{array}$$
- (7)
- a flat surface ${H}^{1}\times {H}^{1}$ in ${H}_{s}^{m}(-1)$ as $\left(\right)$ with $b,c,s>0$ and ${b}^{2}+{c}^{2}\ge 1;$
- (8)
- a flat surface ${H}^{1}\times {H}^{1}$ in ${H}_{s}^{m}(-1)$ as $\left(\right)$ with $b,c>0$, $s\ge 2$ and ${b}^{2}+{c}^{2}<1;$
- (9)
- a flat surface ${H}^{1}\times {S}^{1}$ in ${H}_{s}^{m}(-1)\subset {\mathbb{E}}_{s+1}^{m+1}$ as $(bcoshu,0,\dots ,0,bsinhu,ccosv,csinv,$$\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(\right)open="("\; close=")">{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}}$$
- (12)
- a flat surface immersed in ${H}_{s}^{m}(-1)\subset {\mathbb{E}}_{s+1}^{m+1}$ as$$r\left(\right)open="("\; close=")">{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}}$$
- (13)
- a flat surface immersed in ${H}_{s}^{m}(-1)\subset {\mathbb{E}}_{s+1}^{m+1}$ as$$\begin{array}{c}r\left(\right)open="("\; close=")">{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}\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(\right)open="("\; close=")">{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}\hfill \end{array}$$
- (15)
- a flat surface immersed in ${H}_{s}^{m}(-1)\subset {\mathbb{E}}_{s+1}^{m+1}$ as$$r\left(\right)open="("\; close=")">{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}$$
- (16)
- a flat surface immersed in ${H}_{s}^{m}(-1)\subset {\mathbb{E}}_{s+1}^{m+1}$ as$$r\left(\right)open="("\; close=")">{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}$$
- (17)
- a surface of constant negative curvature immersed in ${H}_{s}^{m}(-1)\subset {\mathbb{E}}_{s+1}^{m+1}$ as$$\begin{array}{c}r\left(\right)open="("\; close>\phantom{\rule{-2.168pt}{0ex}}sinh\left(\frac{2u}{\sqrt{3}}\right)-\frac{{v}^{2}}{3}-\left(\right)open="("\; close=")">\frac{7}{8}+\frac{{v}^{4}}{18}{e}^{\frac{2u}{\sqrt{3}}},\phantom{\rule{0.166667em}{0ex}}v+\left(\right)open="("\; close=")">\frac{{v}^{3}}{3}-\frac{v}{4}\hfill & {e}^{\frac{2u}{\sqrt{3}}},\frac{1}{2}+\phantom{\rule{4.pt}{0ex}}\frac{{v}^{2}}{2}{e}^{\frac{2u}{\sqrt{3}}},\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(\right)open="("\; close>\phantom{\rule{-2.168pt}{0ex}}sinh\left(\frac{2u}{\sqrt{3}}\right)-\frac{{v}^{2}}{3}-\left(\right)open="("\; close=")">\frac{7}{8}+\frac{{v}^{4}}{18}{e}^{\frac{2u}{\sqrt{3}}},\phantom{\rule{0.166667em}{0ex}}v+\left(\right)open="("\; close=")">\frac{{v}^{3}}{3}-\frac{v}{4}\hfill & {e}^{\frac{2u}{\sqrt{3}}},\phantom{\rule{4.pt}{0ex}}\frac{1}{2}+\frac{{v}^{2}}{2}{e}^{\frac{2u}{\sqrt{3}}},\end{array}\hfill \left(\right)open\; close=")">\phantom{\rule{28.90755pt}{0ex}}\frac{\sqrt{1-{r}^{2}}}{r},0,\dots ,0,v+\left(\right)open="("\; close=")">\frac{{v}^{3}}{3}+\frac{v}{4}& {e}^{\frac{2u}{\sqrt{3}}},\phantom{\rule{0.166667em}{0ex}}sinh\left(\frac{2u}{\sqrt{3}}\right)-\frac{{v}^{2}}{3}-\left(\right)open="("\; close=")">\frac{1}{8}+\frac{{v}^{4}}{18}\\ {e}^{\frac{2u}{\sqrt{3}}}$$

- (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 $\left(\right)$;
- (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 $\left(\right)$;
- (5)
- a flat minimal surface in a totally geodesic ${\mathbb{E}}_{1}^{3}\subset {\mathbb{E}}_{s}^{m}$ given by$$\left(\right)open="("\; close=")">\frac{1}{6}{(x-y)}^{3}+x,\frac{1}{6}{(x-y)}^{3}+y,\frac{1}{2}{(x-y)}^{2}$$
- (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 $\left(\right)open="("\; close=")">asinhx,acoshx,bcosy,bsiny$ 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$$\left(\right)open="("\; close=")">\frac{{a}^{2}{x}^{2}}{2},\frac{x}{2}-\frac{{a}^{4}{x}^{2}}{6}+y,\frac{x}{2}+\frac{{a}^{4}{x}^{2}}{6}-y$$
- (9)
- a non-minimal flat surface in a totally geodesic ${\mathbb{E}}_{2}^{3}\subseteq {\mathbb{E}}_{s}^{m}$ defined by$$\begin{array}{c}(\frac{1}{2b}cos\left(\right)open="("\; close=")">\frac{\sqrt{2b}}{a}({a}^{2}x+by),\frac{1}{2b}sin\left(\right)open="("\; close=")">\frac{\sqrt{2b}}{a}({a}^{2}x+by)\hfill & ,\frac{{a}^{2}x-by}{a\sqrt{2b}}),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}a,b0;\end{array}$$
- (10)
- a non-minimal flat surface in a totally geodesic ${\mathbb{E}}_{2}^{3}\subseteq {\mathbb{E}}_{s}^{m}$ defined by$$\begin{array}{c}(\frac{{a}^{2}x+by}{a\sqrt{2b}},\frac{1}{2b}cosh\left(\right)open="("\; close=")">\phantom{\rule{-0.166667em}{0ex}}\frac{\sqrt{2b}}{a}({a}^{2}x-by),\frac{1}{2b}sinh\left(\right)open="("\; close=")">\frac{\sqrt{2b}}{a}({a}^{2}x-by)\hfill & ),\phantom{\rule{0.277778em}{0ex}}a,b0;\end{array}$$
- (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 $\left(\right)$ 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$$\begin{array}{c}\left(\right)open="("\; close=")">cos\left(\right)open="("\; close=")">\frac{\sqrt{b}({a}^{3}x+by)}{{a}^{5/2}}\phantom{\rule{-0.166667em}{0ex}},sin\left(\right)open="("\; close=")">\frac{\sqrt{b}({a}^{3}x+by)}{{a}^{5/2}}\hfill & \phantom{\rule{-0.166667em}{0ex}},cosh\left(\right)open="("\; close=")">\frac{\sqrt{b}({a}^{3}x-by)}{{a}^{5/2}}\\ \phantom{\rule{-0.166667em}{0ex}},sinh\left(\right)open="("\; close=")">\frac{\sqrt{b}({a}^{3}x-by)}{{a}^{5/2}}\end{array},$$
- (15)
- a non-minimal flat surface in a totally geodesic ${\mathbb{E}}_{3}^{4}\subseteq {\mathbb{E}}_{s}^{m}$ defined by$$\begin{array}{c}(\frac{\sqrt[4]{{\delta}^{2}+{\phi}^{2}}cos\left(\right)open="("\; close=")">\lambda \left(\right)open="("\; close=")">bx+\sqrt{{\delta}^{2}+{\phi}^{2}}y}{}\sqrt{2}b\sqrt{\sqrt{{\delta}^{2}+{\phi}^{2}}+\delta}\hfill & ,\frac{\sqrt[4]{{\delta}^{2}+{\phi}^{2}}sin\left(\right)open="("\; close=")">\lambda \left(\right)open="("\; close=")">bx+\sqrt{{\delta}^{2}+{\phi}^{2}}y}{}\\ \sqrt{2}b\sqrt{\sqrt{{\delta}^{2}+{\phi}^{2}}+\delta}\end{array}$$$$\begin{array}{c}\lambda =\frac{\sqrt{b\sqrt{{\delta}^{2}+{\phi}^{2}}+b\delta}}{\sqrt{{\delta}^{2}+{\phi}^{2}}},\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mu =\frac{\sqrt{b\sqrt{{\delta}^{2}+{\phi}^{2}}-b\delta}}{\sqrt{{\delta}^{2}+{\phi}^{2}}},\hfill \end{array}$$

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 $\alpha \left(u\right)=\left(\right)open="("\; close=")">\frac{4}{3}{u}^{3}+u,\frac{4}{3}{u}^{3}-u,2{u}^{2}$.

**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 $\left(\right)open="("\; close=")">{u}^{2}+{v}^{2}+\frac{1}{4},{u}^{2}+{v}^{2}-\frac{1}{4},u,v$
- (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$$\left(\right)open="("\; close=")">u,{u}^{2}+{v}^{2}-\frac{1}{4},{u}^{2}+{v}^{2}+\frac{1}{4},v$$
- (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$$\left(\right)open="("\; close=")">{u}^{2}+{v}^{2}+{a}^{2}+\frac{1}{4},{u}^{2}+{v}^{2}+{a}^{2}-\frac{1}{4},u,v,\sqrt{1+{a}^{2}}$$
- (6)
- a flat surface defined by$$\left(\right)open="("\; close=")">{v}^{2}-\frac{3}{4}+{a}^{2},acosu,asinu,v,{v}^{2}-\frac{5}{4}+{a}^{2}$$
- (7)
- a flat surface defined by$$\frac{1}{\sqrt{1+{a}^{2}}}\left(\right)open="("\; close=")">{u}^{2}+{v}^{2}-\frac{3}{4},{u}^{2}+{v}^{2}-\frac{5}{4},u,v,a$$
- (8)
- a marginally trapped flat surface defined by $\frac{1}{2}\left(\right)open="("\; close=")">2{u}^{2}-1,2{u}^{2}-2,2u,sinv,cosv$
- (9)
- a marginally trapped flat surface defined by$$\left(\right)open="("\; close=")">\frac{b}{\sqrt{4-{b}^{2}}},\frac{cosu}{\sqrt{2-b}},\frac{sinu}{\sqrt{2-b}},\frac{cosv}{\sqrt{2+b}},\frac{sinv}{\sqrt{2+b}}$$
- (10)
- a marginally trapped flat surface defined by$$\left(\right)open="("\; close=")">\frac{coshu}{\sqrt{b-2}},\frac{sinhu}{\sqrt{b-2}},\frac{cosv}{\sqrt{2+b}},\frac{sinv}{\sqrt{2+b}},\frac{b}{\sqrt{{b}^{2}-4}}$$

**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 $\frac{1}{\sqrt{c}}\left(\right)open="("\; close=")">{u}^{2}+{v}^{2}-\frac{3}{4},{u}^{2}+{v}^{2}-\frac{5}{4},u,v$
- (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$$\left(\right)open="("\; close=")">\sqrt{1-{a}^{2}},{u}^{2}+{v}^{2}+{a}^{2}+\frac{1}{4},{u}^{2}+{v}^{2}+{a}^{2}-\frac{1}{4},u,v$$
- (6)
- a flat surface defined by$$\left(\right)open="("\; close=")">a,b\left(\right)open="("\; close=")">{u}^{2}+{v}^{2}-\frac{3}{4},bu,bv$$
- (7)
- a flat surface defined by$$\left(\right)open="("\; close=")">{v}^{2}+\frac{5}{4}-{a}^{2},acoshu,asinhu,v,{v}^{2}+\frac{3}{4}-{a}^{2}$$
- (8)
- the marginally trapped flat surface defined by$$\left(\right)open="("\; close=")">{u}^{2}+1,\frac{1}{2}coshv,u,\frac{1}{2}sinhv,{u}^{2}+\frac{1}{2}$$
- (9)
- a marginally trapped flat surface defined by$$\left(\right)open="("\; close=")">\frac{coshu}{\sqrt{2-b}},\frac{coshv}{\sqrt{2+b}},\frac{sinhu}{\sqrt{2-b}},\frac{sinhv}{\sqrt{2+b}},\frac{b}{\sqrt{4-{b}^{2}}}$$
- (10)
- a flat marginally trapped surface defined by$$\left(\right)open="("\; close=")">\frac{b}{\sqrt{{b}^{2}-4}},\frac{coshv}{\sqrt{b+2}},\frac{sinhu}{\sqrt{b+2}},\frac{cosu}{\sqrt{b-1b}},\frac{sinu}{\sqrt{b-2}}$$

**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 $\left(\right),b$ with ${a}^{2}-{b}^{2}=1;$
- (7)
- a flat surface defined by$$\left(\right)open="("\; close=")">acosv-\frac{a(u-v)}{2}sinv,asinv+\frac{a(u-v)}{2}cosv,\frac{a(u-v)}{2}sinv,\frac{a(u-v)}{2}cosv,b$$
- (8)
- a flat surface defined by$$\left(\right)$$
- (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$$\left(\right)open="("\; close=")">\frac{{b}^{2}({u}^{2}-{k}^{2}-1)-1}{2{b}^{2}k},u,\frac{cosbv}{b},\frac{sinbv}{b},\frac{{b}^{2}({u}^{2}+{k}^{2}-1)-1}{2{b}^{2}k}$$
- (11)
- a surface defined by$$\left(\right)open="("\; close=")">\frac{-{a}^{2}({v}^{2}+{k}^{2}+1)+1}{2{a}^{2}k},\frac{sinhau}{a},\frac{coshau}{a},v,\frac{{a}^{2}({k}^{2}-{v}^{2}-1)-1}{2{a}^{2}k}$$
- (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 $\left(\right)open="("\; close=")">{u}^{2}-{v}^{2}-\frac{5}{4},u,v,{u}^{2}-{v}^{2}-\frac{3}{4}$
- (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$$\left(\right)open="("\; close=")">cosv-\frac{u-v}{2}sinv,sinv+\frac{u-v}{2}cosv,\frac{u-v}{2}sinv,\frac{u-v}{2}cosv$$
- (8)
- the surface defined by$$\left(\right)open="("\; close=")">coshv-\frac{u+v}{2}sinhv,\frac{u+v}{2}coshv,sinhv-\frac{u+v}{2}coshv,\frac{u+v}{2}sinhv$$

## 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$$\begin{array}{c}\left(\right)open="("\; close=")">\sqrt{{a}^{2}+{b}^{2}-1},asinhu,acoshu,bcosv,bsinv,\phantom{\rule{0.277778em}{0ex}}a,b0,{a}^{2}+{b}^{2}\ge 1;\hfill \end{array}$$
- (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 $\left(\right)open="("\; close=")">acosu,asinu,bcosv,bsinv,\sqrt{1+{a}^{2}-{b}^{2}}$
- (5)
- a flat surface defined by$$\begin{array}{c}\left(\right)open="("\; close=")">ku,p{u}^{2}+\frac{(1-{b}^{2})\phi}{{k}^{2}}-\frac{{k}^{2}}{4\phi},bsinv,bcosv,p{u}^{2}+\frac{(1-{b}^{2})\phi}{{k}^{2}}+\frac{{k}^{2}}{4\phi},\phantom{\rule{0.277778em}{0ex}}b,k,p,\phi \ne 0;\hfill \end{array}$$
- (6)
- a flat surface defined by $\left(\right)open="("\; close=")">\sqrt{{b}^{2}-{a}^{2}-1},acoshu,asinhu,bcosv,bsinv$
- (7)
- a flat surface defined by$$\begin{array}{c}\left(\right)open="("\; close=")">p{u}^{2}+\frac{({b}^{2}-1)\phi}{{k}^{2}}+\frac{{k}^{2}}{4\phi},bsinhv,bcoshv,ku,p{u}^{2}+\frac{({b}^{2}-1)\phi}{{k}^{2}}-\frac{{k}^{2}}{4\phi},\phantom{\rule{0.277778em}{0ex}}b,k,p,\phi \ne 0;\hfill \end{array}$$
- (8)
- a flat surface given by $\left(\right)open="("\; close=")">acoshu,bsinhv,asinhu,bcoshv,\sqrt{1+{a}^{2}-{b}^{2}}$
- (9)
- a marginally trapped surface of constant curvature one defined by$$\left(\right)open="("\; close=")">\frac{xy}{x+y},\frac{2}{x+y},\frac{x-y}{x+y},\frac{2+xy}{x+y},0$$
- (10)
- a flat surface defined by $\left(\right)open="("\; close=")">x+xy,y-xy,x-y+xy,1+xy,0$
- (11)
- a surface of positive curvature ${c}^{2}$ defined by$$\left(\right)open="("\; close=")">\frac{xy-{c}^{2}}{{c}^{2}(x+y)},\frac{2\sqrt{1-{c}^{2}}\phantom{\rule{0.166667em}{0ex}}y}{{c}^{2}(x+y)},\frac{xy+{c}^{2}}{{c}^{2}(x+y)},\frac{{c}^{2}(x+y)-2y}{{c}^{2}(x+y)},0$$
- (12)
- a surface of positive curvature ${c}^{2}$ defined by$$\left(\right)open="("\; close=")">0,\frac{xy-{c}^{2}}{{c}^{2}(x+y)},\frac{xy+{c}^{2}}{{c}^{2}(x+y)},\frac{{c}^{2}(x+y)-2y}{{c}^{2}(x+y)},\frac{2\sqrt{{c}^{2}-1}\phantom{\rule{0.166667em}{0ex}}y}{{c}^{2}(x+y)}$$
- (13)
- a surface of negative curvature $-{c}^{2}$ defined by$$\begin{array}{c}\frac{1}{c}\left(\right)open="("\; close=")">coshu-sinhutanhv,sinhutanhv,sinhu-coshutanhv,\sqrt{1+{c}^{2}},0,\phantom{\rule{0.166667em}{0ex}}c0;\hfill \end{array}$$
- (14)
- a flat surface defined by$$\begin{array}{c}(\frac{1+8{c}^{2}+2v}{4c}cosu+\frac{1+v}{2c}sinu,\frac{4{c}^{2}-1}{4c}cosu+\left(\right)open="("\; close=")">c+\frac{v}{2c}sinu,\hfill \end{array}$$
- (15)
- a flat surface defined by$$\begin{array}{c}\left(\right)open="("\; close=")">{e}^{u}-\frac{(2c-v){e}^{-u}}{8c},\frac{v{e}^{u}}{4}-\frac{{e}^{-u}}{2c},{e}^{u}+\frac{(2c-v){e}^{-u}}{8c},\frac{v{e}^{u}}{4}+\frac{{e}^{-u}}{2c},0,\phantom{\rule{0.277778em}{0ex}}c0;\hfill \end{array}$$
- (16)
- a flat surface defined by$$\begin{array}{c}\left(\right)open="("\; close=")">x+\frac{y}{2}+\frac{2{c}^{2}{y}^{3}}{3},xy+\frac{{c}^{2}{y}^{4}}{6},x-\frac{y}{2}+\frac{2{c}^{2}{y}^{3}}{3},c{y}^{2},1+xy+\frac{{c}^{2}{y}^{4}}{6},\phantom{\rule{0.277778em}{0ex}}c0;\hfill \end{array}$$
- (17)
- a flat surface defined by$$\begin{array}{c}\hfill \left(\right)open="("\; close=")">avsinhu+bcoshu,avcoshu,avcoshu+bsinhu,avsinhu,\sqrt{1+{b}^{2}},\phantom{\rule{0.166667em}{0ex}}a,b\ne 0;\end{array}$$
- (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$$\begin{array}{c}\left(\right)open="("\; close=")">vcosu+\frac{sinu}{c},vsinu-\frac{cosu}{c},vcosu-\frac{sinu}{c},vsinu+\frac{cosu}{c},1,\phantom{\rule{0.166667em}{0ex}}c0;\hfill \end{array}$$
- (20)
- a flat surface defined by$$\begin{array}{c}\hfill \left(\right)open="("\; close>cosucosv-\frac{sinusinv}{c},cosusinv+\frac{sinucosv}{c},cosucosv+\frac{sinusinv}{c},\hfill \end{array}\hfill \left(\right)open\; close=")">\phantom{\rule{57.81621pt}{0ex}}cosusinv-\frac{sinucosv}{c},1\phantom{\rule{-0.72229pt}{0ex}},\phantom{\rule{0.277778em}{0ex}}c0;\hfill $$
- (21)
- a flat surface defined by$$\begin{array}{c}\left(\right)open="("\; close=")">{e}^{v}cosu+\frac{{e}^{-v}sinu}{c},{e}^{-v}cosu-\frac{{e}^{v}sinu}{c},{e}^{v}cosu-\frac{{e}^{-v}sinu}{c},{e}^{-v}cosu+\frac{{e}^{v}sinu}{c},1,\phantom{\rule{0.277778em}{0ex}}c0;\hfill \end{array}$$
- (22)
- a flat surface defined by $\left(\right)open="("\; close=")">{e}^{u}+a{e}^{-u}v,{e}^{u}v-a{e}^{-u},{e}^{u}-a{e}^{-u}v,{e}^{u}v+a{e}^{-u},1$
- (23)
- a flat surface defined by $\left(\right)open="("\; close=")">{e}^{u}-a{e}^{-u},{e}^{v}+a{e}^{-v},{e}^{u}+a{e}^{-u},{e}^{v}-a{e}^{-v},1$
- (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$$\begin{array}{c}\frac{1}{\sqrt{2}}\left(\right)open="("\; close=")">\phantom{\rule{-0.166667em}{0ex}}sin\left(\right)open="("\; close=")">ax+\frac{y}{a}\phantom{\rule{-1.4457pt}{0ex}},cos\left(\right)open="("\; close=")">ax+\frac{y}{a}\hfill & \phantom{\rule{-1.4457pt}{0ex}},cosh\left(\right)open="("\; close=")">ax-\frac{y}{a}\\ \phantom{\rule{-1.4457pt}{0ex}},sinh\left(\right)open="("\; close=")">ax-\frac{y}{a}\end{array},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}a0;$$
- (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$$\begin{array}{c}\phantom{\rule{-21.68121pt}{0ex}}\frac{1}{\sqrt{2(1+{b}^{2})}}\left(\right)open="("\; close=")">\sqrt{2}b,cos\left(\right)open="("\; close=")">kx+\frac{{k}^{3}}{{\gamma}^{2}}y,sin\left(\right)open="("\; close=")">kx+\frac{{k}^{3}}{{\gamma}^{2}}y\hfill & ,cosh\left(\right)open="("\; close=")">kx-\frac{{k}^{3}}{{\gamma}^{2}}y\\ ,sinh\left(\right)open="("\; close=")">kx-\frac{{k}^{3}}{{\gamma}^{2}}y\end{array}$$
- (6)
- A non-minimal flat surface given by$$\begin{array}{c}\phantom{\rule{-36.135pt}{0ex}}(\frac{b\phi}{\sqrt{{\delta}^{2}+(1+{b}^{2}){\phi}^{2}}},\frac{\sqrt{\sqrt{1+{b}^{2}}({\delta}^{2}+{\phi}^{2})-b\delta \sqrt{{\delta}^{2}+{\phi}^{2}}}}{\sqrt{2}\sqrt[4]{1+{b}^{2}}\sqrt{{\delta}^{2}+(1+{b}^{2}){\phi}^{2}}}cos\phantom{\rule{-0.166667em}{0ex}}\left(\right)open="("\; close=")">\lambda (\sqrt{1+{b}^{2}}x+\sqrt{{\delta}^{2}+{\phi}^{2}}y,\hfill \end{array}\phantom{\rule{21.68121pt}{0ex}}\frac{\sqrt{\phantom{\rule{-0.166667em}{0ex}}\sqrt{1+{b}^{2}}({\delta}^{2}+{\phi}^{2})+b\delta \sqrt{{\delta}^{2}+{\phi}^{2}}}}{\sqrt{2}\sqrt[4]{1+{b}^{2}}\sqrt{{\delta}^{2}+(1+{b}^{2}){\phi}^{2}}}cosh\phantom{\rule{-0.166667em}{0ex}}\left(\right)open="("\; close=")">\mu (\sqrt{1+{b}^{2}}x-\sqrt{{\delta}^{2}+{\phi}^{2}}y,\hfill $$$$\begin{array}{c}\lambda =\frac{\sqrt{\sqrt{1+{b}^{2}}\sqrt{{\delta}^{2}+{\phi}^{2}}+b\delta}}{\sqrt{{\delta}^{2}+{\phi}^{2}}},\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mu =\frac{\sqrt{\sqrt{1+{b}^{2}}\sqrt{{\delta}^{2}+{\phi}^{2}}-b\delta}}{\sqrt{{\delta}^{2}+{\phi}^{2}}}.\hfill \end{array}$$

## 19. Parallel Lorentz Surfaces in ${\mathbb{C}}_{1}^{2}$, $C{P}_{1}^{2}$ and $C{H}_{1}^{2}$

#### 19.1. Hopf Fibrations

**Remark**

**8.**

#### 19.2. Classification of Parallel Lorentzian Surface in ${\mathbb{C}}_{1}^{2}$

**Theorem**

**44.**

- (1)
- a Lorentzian totally geodesic surface;
- (2)
- a Lorentzian product of parallel curves;
- (3)
- a complex circle, given by $(a+ib)\left(\right)open="("\; close=")">cos(x+iy),sin(x+iy)$ with $a,b\in \mathbf{R},\phantom{\rule{0.277778em}{0ex}}(a,b)\ne (0,0);$
- (4)
- a B-scroll over the null cubic in ${\mathbb{E}}_{1}^{3}\subseteq {\mathbb{C}}_{1}^{2}$;
- (5)
- a B-scroll over the null cubic in ${\mathbb{E}}_{2}^{3}\subseteq {\mathbb{C}}_{1}^{2}$;
- (6)
- a surface given by$$\frac{{e}^{-iy}}{\sqrt{2}}\left(\right)open="("\; close=")">i(1+a)-x-ay,i(1-a)+x+ay$$
- (7)
- a surface with light-like mean curvature vector given by $\left(q\right(x,y),x,y,q(x,y\left)\right)$ with $q(x,y)=a{x}^{2}+bxy+c{y}^{2}+dx+ey+f$ and $a,b,c,d,e,f\in \mathbf{R}$;
- (8)
- a totally umbilical de Sitter space ${S}_{1}^{2}$ in ${\mathbb{E}}_{1}^{3}\subseteq {\mathbb{C}}_{1}^{2}$, given by $a(0,sinhx,coshxcosy,coshxsiny)$ with $a\in \mathbf{R}\backslash \left\{0\right\};$
- (9)
- a totally umbilical anti-de Sitter space ${H}_{1}^{2}$ in ${\mathbb{E}}_{2}^{3}\subseteq {\mathbb{C}}_{1}^{2}$ given by $a(sinx,cosxcoshy,cosxsinhy,0)$ with $a\in \mathbf{R}\backslash \left\{0\right\}.$

#### 19.3. Classification of Parallel Lorentzian Surface in $C{P}_{1}^{2}$

**Lemma**

**5.**

**Theorem**

**45.**

- (I)
- M is an open part of the totally geodesic, Lagrangian surface $R{P}_{1}^{2}\left(1\right)\subseteq C{P}_{1}^{2}\left(4\right)$.
- (II)
- M is flat and the immersion is congruent to $\pi \circ L$, where $\pi :{S}_{2}^{5}\left(1\right)\to C{P}_{1}^{2}\left(4\right)$ is the Hopf-fibration and $L:{M}_{1}^{2}\to {S}_{2}^{5}\left(1\right)\subseteq {\mathbb{C}}_{1}^{3}$ is locally one of the following twelve maps:
- (1)
- $L=\frac{1}{\sqrt{3}}\left(\right)open="("\; close=")">\sqrt{2}{e}^{\frac{i}{2}x}sinh\left(\right)open="("\; close=")">\frac{\sqrt{3}}{2}y,\phantom{\rule{4pt}{0ex}}\sqrt{2}{e}^{\frac{i}{2}x}cosh\left(\right)open="("\; close=")">\frac{\sqrt{3}}{2}y,\phantom{\rule{4pt}{0ex}}{e}^{-ix}};$
- (2)
- $L=\left(\right)open="("\; close=")">\frac{{e}^{\frac{i}{2}(2x+y+\sqrt{1+4a}y)}}{{(1+4a)}^{1/4}},\phantom{\rule{4pt}{0ex}}\frac{{e}^{\frac{i}{2}(2x+y-\sqrt{1+4a}y)}}{{(1+4a)}^{1/4}},\phantom{\rule{4pt}{0ex}}{e}^{iy},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}a-\frac{1}{4};$
- (3)
- $L=(\frac{(2-i{e}^{-\sqrt{4a-1}y}){e}^{ix+\frac{1}{2}(i+\sqrt{4a-1})y}}{2\sqrt[4]{4a-1}},\frac{(2+i{e}^{-\sqrt{4a-1}y}){e}^{ix+\frac{1}{2}(i+\sqrt{4a-1})y}}{2\sqrt[4]{4a-1}},\phantom{\rule{4pt}{0ex}}{e}^{iy}),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}a>\frac{1}{4};$
- (4)
- $L=\frac{1}{\sqrt{2}}\left(\right)open="("\; close=")">{e}^{i(x+\frac{y}{2})}(1+iy),\phantom{\rule{4pt}{0ex}}{e}^{i(x+\frac{y}{2})}(1-iy),\phantom{\rule{4pt}{0ex}}\sqrt{2}{e}^{iy};$
- (5)
- $L=(\frac{\sqrt{a(2-a-b)}\phantom{\rule{0.166667em}{0ex}}{e}^{i(bx+\frac{(1-b)y}{a(2-a-b)})}}{\sqrt{(a-b)(a+2b-2)}},\phantom{\rule{4pt}{0ex}}\frac{\sqrt{b(2-a-b)}{e}^{i(ax+\frac{(1-a)y}{b(2-a-b)})}}{\sqrt{(a-b)(2a+b-2)}},\frac{\sqrt{ab}\phantom{\rule{0.166667em}{0ex}}{e}^{i((2-a-b)x+\frac{a+b-1}{ab}y)}}{\sqrt{(2a+b-2)(a+2b-2)}})$ with $a>b>2-a-b>0$ or $\phantom{\rule{0.277778em}{0ex}}0>a>b>2-a-b;$
- (6)
- $L=(\frac{\sqrt{b(a+b-2)}{e}^{i(ax+\frac{(1-a)y}{b(2-a-b)})}}{\sqrt{(a-b)(2a+b-2)}},\frac{\sqrt{a(a+b-2)}{e}^{i(bx+\frac{(1-b)y}{a(2-a-b)})}}{\sqrt{(a-b)(a+2b-2)}},\frac{\sqrt{ab}\phantom{\rule{0.166667em}{0ex}}{e}^{i((2-a-b)x+\frac{a+b-1}{ab}y)}}{\sqrt{(2a+b-2)(a+2b-2)}})$ with $a>b>0$ and $a+b>2;$
- (7)
- $L=(\frac{\sqrt{-ab}\phantom{\rule{0.166667em}{0ex}}{e}^{i((2-a-b)x+\frac{a+b-1}{ab}y)}}{\sqrt{(2a+b-2)(a+2b-2)}},\phantom{\rule{4pt}{0ex}}\frac{\sqrt{b(2-a-b)}\phantom{\rule{0.166667em}{0ex}}{e}^{i(ax+\frac{(1-a)y}{b(2-a-b)})}}{\sqrt{(a-b)(2a+b-2)}},\frac{\sqrt{a(a+b-2)}{e}^{i(bx+\frac{(1-b)y}{a(2-a-b)})}}{\sqrt{(a-b)(a+2b-2)}}),$ with $a>0>b>2-a-b;$
- (8)
- $L=\left(\right)open="("\; close>\left(\right)open="("\; close=")">\frac{2i\sqrt{(2a-1)(1-a)}}{2-3a}+\frac{2{a}^{2}(a-1)x+(2a-1)y}{2a\sqrt{(2a-1)(1-a)}}{e}^{i(ax+\frac{y}{2a})},$$$\left(\right)open\; close=")">\frac{(2{a}^{2}(a-1)x+(2a-1)y){e}^{i(ax+\frac{y}{2a})}}{2a\sqrt{(2a-1)(1-a)}},\phantom{\rule{4pt}{0ex}}\frac{a\phantom{\rule{0.166667em}{0ex}}{e}^{i(2(1-a)x+\frac{2a-1}{{a}^{2}}y)}}{3a-2},\phantom{\rule{0.277778em}{0ex}}a\in ({\textstyle \frac{1}{2}},1)\backslash \left\{{\textstyle \frac{2}{3}}\right\};$$
- (9)
- $L=\left(\right)open="("\; close>\frac{(2{a}^{2}(a-1)x+(2a-1)y){e}^{i(ax+\frac{y}{2a})}}{2a\sqrt{(2a-1)(a-1)}},\left(\right)open="("\; close=")">\frac{2{a}^{2}(a-1)x+(2a-1)y}{2a\sqrt{(2a-1)(a-1)}}+\frac{2i\sqrt{(2a-1)(a-1)}}{3a-2}$$$\times {e}^{i(ax+\frac{y}{2a})},\left(\right)open\; close=")">\frac{a\phantom{\rule{0.166667em}{0ex}}{e}^{i(2(1-a)x+\frac{2a-1}{{a}^{2}}y)}}{3a-2},\phantom{\rule{0.277778em}{0ex}}a\in \mathbf{R}\backslash ([{\textstyle \frac{1}{2}},1]\cup \left\{0\right\});$$
- (10)
- $L=\frac{{e}^{\frac{i}{12}(8x+9y)}}{24}\left(\right)open="("\; close=")">1+{(8x-9y)}^{2}+432iy,\phantom{\rule{4pt}{0ex}}2(8x-9y+12i),1-{(8x-9y)}^{2}-432iy;$
- (11)
- $L=\left(\right)open="("\; close>\frac{\sqrt{1-a}\phantom{\rule{0.166667em}{0ex}}{e}^{i(ax+\frac{({a}^{2}-{b}^{2}-a)y}{2(a-1)({a}^{2}+{b}^{2})})}}{b\sqrt{2a-1}\sqrt{{(3a-2)}^{2}+{b}^{2}}}(2b(1-2a)cosh\left(\right)open="("\; close=")">bx+\frac{b(2a-1)y}{2(a-1)({a}^{2}+{b}^{2})}$$$+i(3{a}^{2}-{b}^{2}-2a)sinh\left(\right)open="("\; close=")">bx+\frac{b(2a-1)y}{2(a-1)({a}^{2}+{b}^{2})}),$$$$\frac{\sqrt{1-a}\sqrt{{a}^{2}+{b}^{2}}\phantom{\rule{0.166667em}{0ex}}{e}^{i(ax+\frac{({a}^{2}-{b}^{2}-a)y}{2(a-1)({a}^{2}+{b}^{2})})}}{b\sqrt{2a-1}}sinh\left(\right)open="("\; close=")">bx+\frac{b(2a-1)y}{2(a-1)({a}^{2}+{b}^{2})},$$$$\left(\right)open\; close=")">\frac{\sqrt{{a}^{2}+{b}^{2}}\phantom{\rule{0.166667em}{0ex}}{e}^{i(2(1-a)x+\frac{2a-1}{{a}^{2}+{b}^{2}}y)}}{\sqrt{{(3a-2)}^{2}+{b}^{2}}},witha\in ({\textstyle \frac{1}{2}},1)andb\in \mathbf{R}\backslash \left\{0\right\};$$
- (12)
- $L=\left(\right)open="("\; close>\sqrt{\frac{a-1}{2a-1}}\frac{{e}^{i(ax+\frac{({a}^{2}-{b}^{2}-a)y}{2(a-1)({a}^{2}+{b}^{2})})}}{b\sqrt{{(3a-2)}^{2}+{b}^{2}}}(2b(1-2a)sinh\left(\right)open="("\; close=")">bx+\frac{b(2a-1)y}{2(a-1)({a}^{2}+{b}^{2})}$$$+i(3{a}^{2}-{b}^{2}-2a)cosh\left(\right)open="("\; close=")">bx+\frac{b(2a-1)y}{2(a-1)({a}^{2}+{b}^{2})}),$$$$\sqrt{\frac{a-1}{2a-1}}\frac{\sqrt{{a}^{2}+{b}^{2}}\phantom{\rule{0.166667em}{0ex}}{e}^{i(ax+\frac{({a}^{2}-{b}^{2}-a)y}{2(a-1)({a}^{2}+{b}^{2})})}}{b}cosh\left(\right)open="("\; close=")">bx+\frac{b(2a-1)y}{2(a-1)({a}^{2}+{b}^{2})},$$$$\left(\right)open\; close=")">\frac{\sqrt{{a}^{2}+{b}^{2}}\phantom{\rule{0.166667em}{0ex}}{e}^{i(2(1-a)x+\frac{2a-1}{{a}^{2}+{b}^{2}}y)}}{\sqrt{{(3a-2)}^{2}+{b}^{2}}},witha\in \mathbf{R}\backslash [{\textstyle \frac{1}{2}},1]andb\in \mathbf{R}\backslash \left\{0\right\}.$$

#### 19.4. Classification of Parallel Lorentzian Surface in $C{H}_{1}^{2}$

## 20. Parallel Surfaces in Warped Product $I{\times}_{f}{R}^{n}\left(c\right)$

#### 20.1. Basics on Robertson–Walker Space-Times

#### 20.2. Parallel Submanifolds of Robertson–Walker Space-Times

**Theorem**

**46.**

- (a)
- A transverse submanifold lying in a rest space $S\left({t}_{0}\right)$ of ${L}_{1}^{n}(c,f)$ as a parallel submanifold.
- (b)
- An $\mathcal{H}$-submanifold which is locally a warped product $I{\times}_{f}{P}^{k-1}$, where I is an open interval and ${P}^{k-1}$ is a submanifold of ${R}^{n-1}\left(c\right)$. Further,
- (b.1)
- if ${f}^{\prime}\ne 0$ on I, then $I{\times}_{f}{P}^{k-1}$ is totally geodesic in ${L}_{1}^{m}(k,f)$;
- (b.2)
- if ${f}^{\prime}=0$ on I, then ${P}^{k-1}$ is a parallel submanifold of ${R}^{n-1}\left(c\right)$.

## 21. Thurston’s Eight Three-Dimensional Model Geometries

- (1)
- Euclidean geometry ${\mathbb{E}}^{3}$.
- (2)
- Spherical geometry ${S}^{3}$.
- (3)
- Hyperbolic geometry ${H}^{3}$.
- (4)
- The geometry of ${S}^{2}\times \mathbb{R}$.
- (5)
- The geometry of ${H}^{2}\times \mathbb{R}$.
- (6)
- The geometry $\tilde{S{L}_{2}}\left(\mathbb{R}\right)$. The 3-dimensional Lie group of all $2\times 2$ real matrices with determinant one is denoted by $S{L}_{2}\left(\mathbb{R}\right)$; and $\tilde{S{L}_{2}}\left(\mathbb{R}\right)$ denotes its universal covering. $\tilde{S{L}_{2}}\left(\mathbb{R}\right)$ is a unimodular Lie group with a special left invariant metric. Examples of these manifolds in this geometry include the manifold of unit vectors of the tangent bundle of a hyperbolic surface and, more generally, the Brieskorn homology spheres.
- (7)
- Nil geometry $Ni{l}_{3}$. The group $Ni{l}_{3}$ is a 3-dimensional unimodular Lie group with a special left invariant metric consisting of real matrices of the form$$\begin{array}{c}\hfill \left(\begin{array}{ccc}1\phantom{\rule{0.277778em}{0ex}}& x\phantom{\rule{0.277778em}{0ex}}& y\\ 0& 1& z\\ 0& 0& 1\end{array}\right)\end{array}$$
- (8)
- Sol geometry $So{l}_{3}$. This group $So{l}_{3}$ has the least symmetry of all the eight geometries as the identity component of the stabilizer of a point is trivial.

## 22. Parallel Surfaces in Three-Dimensional Lie Groups

#### 22.1. Milnor’s Classification of 3-Dimensional Unimodular Lie Groups

- (a)
- $\left(\right)open="\langle "\; close="\rangle ">X,X\times Y=0$,
- (b)
- ${|X\times Y|}^{2}=\left(\right)open="\langle "\; close="\rangle ">X,X-{\left(\right)}^{X}2$,
- (c)
- if X and Y are linearly independent, then $det(X,Y,X\times Y)>0,$

**Theorem**

**47.**

#### 22.2. Parallel Surfaces in the Motion Group $E(1,1)$

**Theorem**

**48.**

#### 22.3. Parallel Surfaces in $So{l}_{3}$

**Theorem**

**49.**

- (a)
- an integral surface of the distribution spanned by $\left(\right)$,
- (b)
- an integral surface of the distribution spanned by $\left(\right)$ or $\left(\right)$,

#### 22.4. Parallel Surfaces in the Motion Group $E\left(2\right)$

**Proposition**

**1.**

**Theorem**

**50.**

#### 22.5. Parallel Surfaces in $SU\left(2\right)$

**Proposition**

**2.**

**Proposition**

**3.**

**Theorem**

**51.**

#### 22.6. Parallel Surfaces in the Real Special Linear Group $Sl(2,\mathbb{R})$

**Proposition**

**4.**

**Theorem**

**52.**

**Theorem**

**53.**

#### 22.7. Parallel Surfaces in Non-Unimodular Three-Dimensional Lie Groups

**Proposition**

**5.**

**Theorem**

**54.**

- (1)
- Integral surfaces of the distributions spanned by $\{{e}_{1},{e}_{2}\}$, respectively $\{{e}_{1},{e}_{3}\}$. These surfaces are totally geodesic and of constant negative curvature $-{(1+\xi )}^{2}$, respectively $-{(1-\xi )}^{2}$.
- (2)
- Integral surfaces of the distribution spanned by $\{{e}_{2},{e}_{3}\}$. These surfaces are flat and of constant mean curvature 1.

#### 22.8. Parallel Surfaces in the Heisenberg Group $Ni{l}_{3}$

**Theorem**

**55.**

**Remark**

**9.**

## 23. Parallel Surfaces in Three-Dimensional Lorentzian Lie Groups

#### 23.1. Three-Dimensional Lorentzian Lie Groups

**Theorem**

**56.**

- (1)
- Type ${\mathfrak{g}}_{1}$:$$\left(\right)open="["\; close="]">{e}_{1},{e}_{2}=-\alpha {e}_{1}-\beta {e}_{2},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\left(\right)open="["\; close="]">{e}_{2},{e}_{3}$$
- (2)
- Type ${\mathfrak{g}}_{2}$:$$\left(\right)open="["\; close="]">{e}_{1},{e}_{2}=-\beta {e}_{2}+\gamma {e}_{3},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\left(\right)open="["\; close="]">{e}_{2},{e}_{3}$$
- (3)
- Type ${\mathfrak{g}}_{3}$:$$\left(\right)open="["\; close="]">{e}_{1},{e}_{2}=-\beta {e}_{2},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\left(\right)open="["\; close="]">{e}_{2},{e}_{3}$$
- (4)
- Type ${\mathfrak{g}}_{4}$:$$\left(\right)open="["\; close="]">{e}_{1},{e}_{2}=-\beta {e}_{2}+{e}_{3},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\left(\right)open="["\; close="]">{e}_{2},{e}_{3}$$
- (5)
- Type ${\mathfrak{g}}_{5}$:$$\left(\right)open="["\; close="]">{e}_{1},{e}_{2}=\alpha {e}_{1}+\beta {e}_{2},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\left(\right)open="["\; close="]">{e}_{2},{e}_{3}$$
- (6)
- Type ${\mathfrak{g}}_{6}$:$$\left(\right)open="["\; close="]">{e}_{1},{e}_{2}=\gamma {e}_{2}+\delta {e}_{3},\phantom{\rule{4pt}{0ex}}\left(\right)open="["\; close="]">{e}_{2},{e}_{3}$$
- (7)
- Type ${\mathfrak{g}}_{7}$:$$\left(\right)open="["\; close="]">{e}_{1},{e}_{2}=\alpha {e}_{1}+\beta {e}_{2}+\beta {e}_{3},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\left(\right)open="["\; close="]">{e}_{2},{e}_{3}$$

#### 23.2. Classification of Parallel Surfaces in Three-Dimensional Lorentzian Lie Groups

**Theorem**

**57.**

**Theorem**

**58.**

- (a)
- M is an integral surface of the distribution spanned by $\{{e}_{2},{e}_{3}\}$. This case only occurs if $\alpha =0$ and M is parallel, flat and minimal but not totally geodesic.
- (b)
- M is an integral surface of the distribution spanned by $\{{e}_{1},c{e}_{2}+b{e}_{3}\}$, where b and c are real constants satisfying ${b}^{2}-{c}^{2}=\epsilon =\pm 1,\phantom{\rule{0.166667em}{0ex}}bc=-\epsilon \beta /\left(2\gamma \right).$ This case only occurs if $\alpha =2\beta $ and M is totally geodesic.

**Theorem**

**59.**

- (a)
- M is an integral surface of the distribution spanned by $\{{e}_{2},{e}_{3}\}$. This case only occurs if $\gamma =0$ and M is flat and minimal but not totally geodesic.
- (b)
- M is an integral surface of the distribution spanned by $\{{e}_{2},{e}_{3}\}$. This case only occurs if $\alpha =0$ and M is flat and minimal but not totally geodesic.
- (c)
- M is an integral surface of the distribution spanned by $\{{e}_{1},{e}_{3}\}$. This case only occurs if $\beta =0$ and M is flat and minimal but not totally geodesic.
- (d)
- M is an integral surface of the distribution spanned by $\{{E}_{1}={e}_{1},{E}_{2}=c{e}_{2}+b{e}_{3}\}$, where b and c are functions on M satisfying ${b}^{2}-{c}^{2}=\epsilon $ and ${E}_{1}b=\beta c,\phantom{\rule{0.166667em}{0ex}}{E}_{1}c=\beta b,\phantom{\rule{0.166667em}{0ex}}{E}_{2}b={k}_{1}\epsilon c,\phantom{\rule{0.166667em}{0ex}}{E}_{2}c={k}_{1}\epsilon b,$ for some real constant ${k}_{1}$. This case only occurs if $\beta =\gamma $ and M is flat.
- (e)
- M is an integral surface of the distribution spanned by $\{c{e}_{2}+b{e}_{3},{e}_{1}\}$. Here, b and c are real constants satisfying ${b}^{2}=\gamma \epsilon /(\gamma -\beta ),\phantom{\rule{0.166667em}{0ex}}{c}^{2}=\beta \epsilon /(\gamma -\beta ).$ This case only occurs if $\alpha =\beta +\gamma $ and $\beta \ne \gamma $ and M is totally geodesic.
- (f)
- M is an integral surface of the distribution spanned by $\{{E}_{1}=c{e}_{1}+a{e}_{3},{E}_{2}={e}_{2}\}$, where a and c are functions on the surface satisfying ${a}^{2}-{c}^{2}=\epsilon $ and ${E}_{1}a={k}_{2}\epsilon c,\phantom{\rule{0.166667em}{0ex}}{E}_{1}c={k}_{2}\epsilon a,\phantom{\rule{0.166667em}{0ex}}{E}_{2}a=-\alpha c,\phantom{\rule{0.166667em}{0ex}}{E}_{2}c=-\alpha a,$ for some real constant ${k}_{2}$. This case only occurs if $\alpha =\gamma $ and M is flat.
- (g)
- M is an integral surface of the distribution spanned by $\{c{e}_{1}+a{e}_{3},{e}_{2}\}$. Here, a and c are real constants satisfying ${a}^{2}=-\gamma \epsilon /(\alpha -\gamma ),\phantom{\rule{0.166667em}{0ex}}{c}^{2}=-\alpha \epsilon /(\alpha -\gamma ).$ This case only occurs if $\beta =\alpha +\gamma $ and $\alpha \ne \gamma $ and M is totally geodesic.
- (h)
- M is an integral surface of the distribution spanned by $\{{E}_{1}=b{e}_{1}-a{e}_{2},{E}_{2}={e}_{3}\}$, where a and b are functions satisfying ${a}^{2}+{b}^{2}=1$ and$${E}_{1}a=\frac{{k}_{3}b}{{b}^{2}-{a}^{2}},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{E}_{1}b=-\frac{{k}_{3}a}{{b}^{2}-{a}^{2}},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{E}_{2}a=\frac{b\alpha}{{b}^{2}-{a}^{2}},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{E}_{2}b=-\frac{a\alpha}{{b}^{2}-{a}^{2}},$$
- (i)
- M is an integral surface of the distribution spanned by $\{b{e}_{1}-a{e}_{2},{e}_{3}\}$, where a and b are constants satisfying ${a}^{2}=-\beta /(\alpha -\beta ),\phantom{\rule{0.166667em}{0ex}}{b}^{2}=\alpha /(\alpha -\beta ).$ This case only occurs if $\gamma =\alpha +\beta $ and $\alpha \ne \beta $ and M is totally geodesic.

**Theorem**

**60.**

- (a)
- M is an integral surface of the distribution spanned by $\{{e}_{2},{e}_{3}\}$. This case only occurs if $\alpha =0$. M is parallel, flat and minimal but not totally geodesic.
- (b)
- M is an integral surface of the distribution spanned by $\{{e}_{1},c{e}_{2}+b{e}_{3}\}$, where b and c are constants satisfying ${b}^{2}-{c}^{2}=\epsilon $ and $\beta {b}^{2}+2bc+(\beta -2\eta ){c}^{2}=0$. M is totally geodesic and has constant Gaussian curvature $G=-\epsilon (\beta -\eta )$.

**Theorem**

**61.**

- (a)
- M is an integral surface of the distribution spanned by ${e}_{1}$ and ${e}_{2}$. M is flat but not totally geodesic.
- (b)
- M is an integral surface of the distribution spanned by ${e}_{2}$ and ${e}_{3}$. This case only occurs if either $\beta =\gamma =0$ or $\gamma =\delta =0$. In the first case, M is totally geodesic and has constant Gaussian curvature $K=-{\delta}^{2}\le 0$. In the second case, M is flat and minimal but not necessarily totally geodesic.
- (c)
- M is an integral surface of the distribution spanned by ${e}_{1}$ and ${e}_{3}$. This case only occurs if either $\alpha =\beta =0$ or $\beta =\gamma =0$. In the first case, M is flat and minimal but not necessarily totally geodesic. In the second case, M is totally geodesic and has constant Gaussian curvature $K={\alpha}^{2}\ge 0$.
- (d)
- M is an integral surface of the distribution spanned by $\{{E}_{1}={e}_{1},{E}_{2}=c{e}_{2}+b{e}_{3}\}$, where b and c are functions satisfying ${b}^{2}-{c}^{2}=\epsilon $ and ${E}_{1}b={E}_{1}c=0,\phantom{\rule{0.166667em}{0ex}}{E}_{2}b=c({k}_{1}-c\delta ),\phantom{\rule{0.166667em}{0ex}}{E}_{2}c=b({k}_{1}-c\delta ),$ for some real constant ${k}_{1}$. This case only occurs if $\alpha =\beta =0$ and M is flat.
- (e)
- M is an integral surface of the distribution spanned by $\{{E}_{1}=c{e}_{1}+a{e}_{3},{E}_{2}={e}_{2}\}$, where a and c are functions satisfying ${a}^{2}-{c}^{2}=\epsilon $ and ${E}_{1}a=-\epsilon c({a}^{2}c\alpha -{k}_{2}),\phantom{\rule{0.166667em}{0ex}}{E}_{1}c=-\epsilon a({a}^{2}c\alpha -{k}_{2}),\phantom{\rule{0.166667em}{0ex}}{E}_{2}a={E}_{2}c=0,$ for some real constant ${k}_{2}$. This case only occurs if $\gamma =\delta =0$ and M is flat.

**Theorem**

**62.**

- (a)
- M is an integral surface of the distribution spanned by ${e}_{1}$ and ${e}_{2}$. This case only occurs if either $\alpha =\beta =0$ or $\beta =\gamma =0$. In the first case, M is parallel, flat and minimal but not necessarily totally geodesic. In the second case, M is totally geodesic.
- (b)
- M is an integral surface of the distribution spanned by ${e}_{2}$ and ${e}_{3}$. M is parallel and flat but not necessarily totally geodesic.
- (c)
- M is an integral surface of the distribution spanned by ${e}_{1}$ and ${e}_{3}$. This case only occurs if either $\beta =\gamma =0$ or $\gamma =\delta =0$. In the first case, M is totally geodesic. In the second case, M is parallel, flat and minimal but not necessarily totally geodesic.
- (d)
- M is an integral surface of the distribution spanned by $\{{E}_{1}=c{e}_{1}+a{e}_{3},{E}_{2}={e}_{2}\}$, where a and c are functions satisfying ${a}^{2}-{c}^{2}=\epsilon $ and ${E}_{1}a=c({k}_{1}-\delta a),\phantom{\rule{0.166667em}{0ex}}{E}_{1}c=a({k}_{1}-\delta a),\phantom{\rule{0.166667em}{0ex}}{E}_{2}a={E}_{2}c=0$ for some real constant ${k}_{1}$. This case only occurs if $\alpha =\beta =0$ and M is parallel and flat.
- (e)
- M is an integral surface of the distribution spanned by $\{{E}_{1}=b{e}_{1}-a{e}_{2},{E}_{2}={e}_{3}\}$, where a and b are functions satisfying ${a}^{2}+{b}^{2}=1$ and ${E}_{1}a=b({k}_{2}+\alpha a),\phantom{\rule{0.166667em}{0ex}}{E}_{1}b=-a({k}_{2}+\alpha b),\phantom{\rule{0.166667em}{0ex}}{E}_{2}a={E}_{2}c=0$ for some real constant ${k}_{2}$. This case only occurs if $\gamma =\delta =0$ and M is parallel and flat.

**Theorem**

**63.**

- (a)
- M is an integral surface of the distribution spanned by $\{{e}_{2},{e}_{3}\}$. This case only occurs if either $\beta =\gamma =0$ or $\gamma =\delta =0$. In the first case, M is totally geodesic. In the second case, M is parallel and flat but not necessarily totally geodesic.
- (b)
- M is an integral surface of the distribution spanned by $\{{E}_{1}={e}_{1},{E}_{2}=c{e}_{2}+b{e}_{3}\}$, where b and c are functions satisfying ${b}^{2}-{c}^{2}=\epsilon $ and ${E}_{1}b={E}_{1}c=0,\phantom{\rule{0.166667em}{0ex}}{E}_{2}b=c((b-c)\delta -{k}_{1}),\phantom{\rule{0.166667em}{0ex}}{E}_{2}c=b((b-c)\delta -{k}_{1})$ for some real constant ${k}_{1}$. This case only occurs if $\alpha =\beta =0$. M is flat but not necessarily totally geodesic.
- (c)
- M is an integral surface of the distribution spanned by ${E}_{1}=(b{e}_{1}-{e}_{2})/\sqrt{1+{b}^{2}}$ and ${E}_{2}=(b{e}_{1}+{b}^{2}{e}_{2}+(1+{b}^{2}){e}_{3})/\sqrt{1+{b}^{2}}$, where b is a function satisfying ${E}_{1}\left(b\right)={E}_{2}\left(b\right)$ and$$\begin{array}{c}\hfill {E}_{1}\left(\right)open="("\; close=")">\frac{{E}_{1}b}{\sqrt{1+{b}^{2}}}+\frac{b(\alpha -\delta )}{1+{b}^{2}}+2\left(\right)open="("\; close=")">\frac{{E}_{1}b}{\sqrt{1+{b}^{2}}}+\frac{b(\alpha -\delta )}{1+{b}^{2}}& \left(\right)open="("\; close=")">\frac{b{E}_{1}b}{\sqrt{1+{b}^{2}}}-\frac{\delta}{\sqrt{1+{b}^{2}}}\\ =0.\end{array}$$

## 24. Parallel Surfaces in Reducible Three-Spaces

#### 24.1. Classification of Parallel Surfaces in Reducible Three-Spaces

**Theorem**

**64.**

- (1)
- M is isometric to an open portion of a surface of type ${\mathbb{M}}^{2}\times \left\{{t}_{0}\right\}$ for some ${t}_{0}\in \mathbf{R}$;
- (2)
- M is isometric to an open portion of a surface of type $\gamma \times {\mathbb{E}}^{1}$, where γ is a curve of constant geodesic curvature in M;
- (3)
- ${\mathbb{M}}^{2}\times {\mathbb{E}}^{1}$ is flat and M is isometric to an open portion of a standard sphere ${S}^{2}\subset {\mathbb{E}}^{3}$.

**Corollary**

**8.**

**Theorem**

**65.**

- (1)
- M is isometric to an open portion of a surface of type ${\mathbb{M}}_{1}^{2}\times \left\{{t}_{0}\right\}$ (respectively ${\mathbb{M}}^{2}\times \left\{{t}_{0}\right\}$) for some real number ${t}_{0}$.
- (2)
- M is isometric to an open portion of a surface of type $\gamma \times {\mathbb{E}}^{1}$ (respectively $\gamma \times {\mathbb{E}}_{1}^{1}$) where γ is a non-degenerate curve of constant geodesic curvature in ${\mathbb{M}}_{1}^{2}$ (respectively ${\mathbb{M}}^{2}$).
- (3)
- The ambient space is flat and M is isometric to an open portion of one of the following surfaces: (a) a hyperbolic plane ${H}^{2}$; (b) an indefinite sphere ${S}_{1}^{2}$; (c) the null scroll ${N}_{1}^{2}$.

**Corollary**

**9.**

#### 24.2. Parallel Surfaces in Walker Three-Manifolds

## 25. Bianchi–Cartan–Vranceanu Spaces

#### 25.1. Basics on Bianchi–Cartan–Vranceanu Spaces

- (1)
- If $\lambda =\mu =0$, it is the Euclidean 3-space.
- (2)
- If $\lambda =0$, $\mu \ne 0$, it is the product of real line and a surface of constant curvature $4\lambda $.
- (3)
- If $\lambda \ne 0$, ${\lambda}^{2}=4\mu $, it is a space of positive constant curvature.
- (4)
- If $\lambda \ne 0$, $\mu >0$, it is $SU\left(2\right)\backslash \{\infty \}$.
- (5)
- If $\lambda \ne 0$, $\mu <0$, it is $\tilde{S{L}_{2}}\left(\mathbb{R}\right)$ with a left-invariant metric.
- (6)
- If $\lambda \ne 0$, $\mu =0$, it is the Heisenberg group $Ni{l}_{3}$ with a left-invariant metric.

#### 25.2. B-Scrolls

#### 25.3. Parallel Surfaces in Bianchi–Cartan–Vranceanu Spaces

**Theorem**

**66.**

- (1)
- If $\lambda \ne 0$, then the only parallel surfaces in ${\tilde{\mathcal{M}}}^{3}(\lambda ,\mu )$ are Hopf cylinders over curves with constant curvature in ${\tilde{\mathcal{M}}}^{2}\left(\mu \right)$.
- (2)
- If $\lambda =0$, then the only parallel surfaces in ${\tilde{\mathcal{M}}}^{3}(\lambda ,\mu )$ with $\mu \ne 0$ are totally geodesic leaves and Hopf cylinders over circles with constant geodesic curvature in ${\tilde{\mathcal{M}}}^{2}\left(\mu \right)$.

## 26. Parallel Surfaces in Homogeneous Three-Spaces

#### 26.1. Homogeneous Three-Spaces

**Theorem**

**67.**

- (i)
- if $dimI\left({M}^{3}\right)=6$, then ${M}^{3}$ is a real space form of constant sectional curvature c, that is, Euclidean space ${\mathbb{E}}^{3}$, hyperbolic space ${H}^{3}\left(c\right)$ or a three-sphere ${S}^{3}\left(c\right)$,
- (ii)
- if $dimI\left({M}^{3}\right)=4$, then ${M}^{3}$ is a Bianchi–Cartan–Vranceanu space (different from ${\mathbb{E}}^{3}$ and ${S}^{3}\left(c\right)$), that is, a Riemannian product ${H}^{2}\left(c\right)\times \mathbb{R}$ or ${S}^{2}\left(c\right)\times \mathbb{R}$ or one of following Lie groups, equipped with a left-invariant metric yielding a four-dimensional isometry group: the special unitary group $SU\left(2\right)$, the universal covering of the special linear group $\tilde{SL}(2,\mathbb{R})$ or the Heisenberg group ${\mathrm{Nil}}_{3}$,
- (iii)
- if $dimI\left({M}^{3}\right)=3$, then ${M}^{3}$ is a general three-dimensional Lie group with left-invariant metric.

#### 26.2. Classification of Parallel Surfaces in Homogeneous Three-Spaces

**Theorem**

**68.**

- (1)
- a real space form ${S}^{3},{\mathbb{E}}^{3}$ or ${H}^{3}$,
- (2)
- a Riemannian product space ${S}^{2}\times \mathbb{R}$ or ${H}^{2}\times \mathbb{R}$,
- (3)
- $SL(2,\mathbb{R})$ with a left-invariant metric determined by the condition ${c}_{2}={c}_{1}+{c}_{3}$ or equivalently ${\mu}_{2}=0$,
- (4)
- the Minkowski motion group $E(1,1)$ with Riemannian 4-symmetric metric, including the model space $So{l}_{3}$,
- (5)
- a non-unimodular Lie group with structure constants $(\xi ,\eta )$ satisfying $\xi \notin \{0,1\}$ and $\eta =0$.

**Theorem**

**69.**

- (1)
- a real space form ${S}^{3},{\mathbb{E}}^{3}$ or ${H}^{3}$,
- (2)
- a Bianchi–Cartan–Vranceanu space,
- (3)
- the Minkowski motion group E(1, 1) with any left-invariant metric, including the model space $So{l}_{3}$,
- (4)
- the Euclidean motion group $E\left(2\right)$ with any left-invariant metric,
- (5)
- a non-unimodular Lie group with structure constants $(\xi ,\eta )$ satisfying $\xi \notin \{0,1\}$.

## 27. Parallel Surfaces in Symmetric Lorentzian Three-Spaces

#### 27.1. Symmetric Lorentzian Three-Spaces

- (I)
- For Segre type $\{11,1\}$, $(N,g)$ is symmetric if and only if
- (i)
- $a=b=c$. Then $(N,g)$ is an Einstein manifold and hence it has constant sectional curvature. If N is connected and simply connected, then $(N,g)$ is isometric to one of the Lorentzian space forms: either ${S}_{1}^{3}$, $\phantom{\rule{0.166667em}{0ex}}{\mathbb{R}}_{1}^{3}$ or ${H}_{1}^{3}$.
- (ii)
- $a=b\ne c$. Then N is reducible as a direct product ${M}^{2}\times {\mathbb{R}}^{1}$, where ${M}^{2}$ is a Riemannian surface of constant curvature. If N is connected and simply connected, $(N,g)$ is then isometric to either ${S}^{2}\times \mathbb{R}$ or ${H}^{2}\times \mathbb{R}$.
- (iii)
- $a\ne b=c$. Then N is reducible as a direct product $\mathbb{R}\times {M}_{1}^{2}$, where ${M}_{1}^{2}$ is a Lorentzian surface of constant sectional curvature. When N is connected and simply connected, $(N,g)$ is isometric to either $\mathbb{R}\times {S}_{1}^{2}$ or $\mathbb{R}\times {H}_{1}^{2}$.

- (II)
- For Segre type $\left\{21\right\}$, $(N,g)$ is symmetric if and only if $a-b=\eta $ and, with respect to a suitable pseudo-orthonormal frame field $\{{e}_{1},{e}_{2},{e}_{3}\}$, the Levi Civita connection of $(N,g)$ is completely described by$$\begin{array}{ccc}{\nabla}_{{e}_{1}}{e}_{1}=A{e}_{2}-A{e}_{3},\hfill & {\nabla}_{{e}_{2}}{e}_{1}=B{e}_{2}-B{e}_{3},\hfill & {\nabla}_{{e}_{3}}{e}_{1}=C{e}_{2}-C{e}_{3},\hfill \\ {\nabla}_{{e}_{1}}{e}_{2}=-A{e}_{1},\hfill & {\nabla}_{{e}_{2}}{e}_{2}=-B{e}_{1},\hfill & {\nabla}_{{e}_{3}}{e}_{2}=-C{e}_{1},\hfill \\ {\nabla}_{{e}_{1}}{e}_{3}=-A{e}_{1},\hfill & {\nabla}_{{e}_{2}}{e}_{3}=-B{e}_{1},\hfill & {\nabla}_{{e}_{3}}{e}_{3}=-C{e}_{1},\hfill \end{array}$$$$g=\left(\right)open="("\; close=")">\begin{array}{ccc}0& 0& 1\\ 0& \epsilon & 0\\ 1& 0& f\end{array},$$$$f(x,y)={x}^{2}\alpha +x\beta \left(y\right)+\xi \left(y\right),$$
- (III)
- For either Segre type $\left\{1z\overline{z}\right\}$ or Segre type $\left\{3\right\}$, $(N,g)$ is never symmetric.

**Theorem**

**70.**

- (i)
- a Lorentzian space form ${S}_{1}^{3}$, ${\mathbb{R}}_{1}^{3}$ or ${H}_{1}^{3}$ or
- (ii)
- a direct product $\mathbb{R}\times {S}_{1}^{2}$, $\mathbb{R}\times {H}_{1}^{2}$, ${S}^{2}\times {\mathbb{R}}_{1}^{1}$ or ${H}^{2}\times {\mathbb{R}}_{1}^{1}$ or
- (iii)
- a space with a Lorentzian metric g locally described by (7)–(8).

#### 27.2. Classification of Parallel Surfaces in Symmetric Lorentzian Three-Spaces

**Theorem**

**71.**

**Theorem**

**72.**

**Remark**

**10.**

## 28. Three Natural Extensions of Parallel Submanifolds

#### 28.1. Submanifolds with Parallel Mean Curvature Vector

#### 28.2. Higher Order Parallel Submanifolds

#### 28.3. Semi-Parallel Submanifolds

## Funding

## Acknowledgments

## Conflicts of Interest

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$({\mathit{c}}_{1},{\mathit{c}}_{2},{\mathit{c}}_{3})$ | Simply-Connected Lie Group | Property |
---|---|---|

$(+,+,+)$ | $SU\left(2\right)$ | Compact and simple |

$(+,+,-)$ | $\tilde{SL}(2,\mathbb{R})$ | Non-compact and simple |

$(+,+,0)$ | $\tilde{E}\left(2\right)$ | Solvable |

$(+,-,0)$ | $E(1,1)$ | Solvable |

$(+,0,0)$ | Heisenberg group | Nilpotent |

$(0,0,0)$ | $({\mathbb{E}}^{3},+)$ | Abelian |

G | $\mathit{\alpha}$ | $\mathit{\beta}$ | $\mathit{\gamma}$ |
---|---|---|---|

$O(1,2)$ or $SL(2,\mathbb{R})$ | + | + | + |

$O(1,2)$ or $SL(2,\mathbb{R})$ | + | − | − |

$SO\left(3\right)$ or $SU\left(2\right)$ | + | + | − |

$E\left(2\right)$ | + | + | 0 |

$E\left(2\right)$ | + | 0 | − |

$E(1,1)$ | + | − | 0 |

$E(1,1)$ | + | 0 | + |

${H}_{3}$ | + | 0 | 0 |

${H}_{3}$ | 0 | 0 | − |

$\mathbb{R}\oplus \mathbb{R}\oplus \mathbb{R}$ | 0 | 0 | 0 |

G | $\mathit{\alpha}$ | $\mathit{\beta}$ |
---|---|---|

$O(1,2)$ or $SL(2,\mathbb{R})$ | $\ne 0$ | $\ne \eta $ |

$E(1,1)$ | 0 | $\ne \eta $ |

$E(1,1)$ | $<0$ | $\eta $ |

$E\left(2\right)$ | $>0$ | $\eta $ |

${H}_{3}$ | 0 | $\eta $ |

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