# Topological Analysis of Fibrations in Multidimensional (C, R) Space

## Abstract

**:**

## 1. Introduction

#### 1.1. Topological Fiber Spaces

#### 1.2. Manifolds and Immersions

#### 1.3. Motivation and Contributions

## 2. Preliminary Concepts

## 3. Topological Fiber Space and Fibrations

#### 3.1. Projective Base in $(C,R)$ Tangent Space

#### 3.2. Topological Fiber Bundle in $(C,R)$ Space

**Remark**

**1.**

#### 3.3. Fiber Space

**Proposition**

**1.**

**Proof.**

**Remark**

**2.**

#### 3.4. Translation in Fiber Space

**Remark**

**3.**

**Proposition**

**2.**

**Proof.**

#### 3.5. Contact Category Fiber

#### 3.6. Oriented Singularities of Function

**Remark**

**4.**

## 4. Algebraic and Topological Properties

**Theorem**

**1.**

**Proof.**

**Corollary**

**1.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Lemma**

**1.**

**Proof.**

**Remark**

**5.**

**Theorem**

**4.**

**Proof.**

**Remark**

**6.**

## 5. Expansion and Singularity

## 6. Comparative Analysis

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Vincze, C. On geometric vector fields of Minkowski spaces and their applications. Differ. Geom. Its Appl.
**2006**, 24, 1–20. [Google Scholar] [CrossRef] [Green Version] - Cleszko, M. Application of Minkowski space to description of anisotropic pore space structure in porous materials. Suppl. Minisymp. Math. Mech.
**2011**, 80, 129–132. [Google Scholar] [CrossRef] - Ersoy, S.; Bilgin, M.; Ince, I. Generalized open sets of Minkowski space. Math. Moravica
**2015**, 19, 49–56. [Google Scholar] [CrossRef] [Green Version] - Agrawal, G.; Shrivastava, S. t-topology on the n-dimensional Minkowski space. J. Math. Phys.
**2009**, 50, 1–6. [Google Scholar] [CrossRef] - Guler, E. Helical hypersurfaces in Minkowski geometry E
_{1}^{4}. Symmetry**2020**, 12, 1206. [Google Scholar] [CrossRef] - Wall, C.T.C. On bundles over a sphere with fiber Euclidean space. Fundam. Math.
**1967**, 61, 57–72. [Google Scholar] [CrossRef] [Green Version] - Preaux, J.P. A survey on Seifert fiber space theorem. ISRN Geom.
**2014**, 2014, 694106. [Google Scholar] [CrossRef] - Wilkes, G. Profinite rigidity for Seifert fiber spaces. Geom. Dedicata
**2017**, 188, 141–163. [Google Scholar] [CrossRef] [Green Version] - Soma, T. Scott’s rigidity theorem for Seifert fibered spaces; Revisited. Trans. Am. Math. Soc.
**2006**, 358, 4057–4070. [Google Scholar] [CrossRef] [Green Version] - Soma, T. A rigidity theorem for Haken manifolds. Math. Proc. Camb. Philos. Soc.
**1995**, 118, 141–160. [Google Scholar] [CrossRef] - Navarro, M.; Palmas, O.; Solis, D.A. On the geometry of null hypersurfaces in Minkowski space. J. Geom. Phys.
**2014**, 75, 199–212. [Google Scholar] [CrossRef] - Kulkarni, R.S.; Raymond, F. 3-dimensional Lorentz space-forms and Seifert fiber spaces. J. Differ. Geom.
**1985**, 21, 231–268. [Google Scholar] [CrossRef] - Lashof, R.; Smale, S. On the immersion of manifolds in Euclidean space. Ann. Math.
**1958**, 68, 562–583. [Google Scholar] [CrossRef] - Butler, D.M.; Pendley, M.H. A visualization model based on the mathematics of fiber bundles. Comput. Phys.
**1989**, 3, 45–51. [Google Scholar] [CrossRef] [Green Version] - Hu, S.T. The equivalence of fiber bundles. Ann. Math.
**1951**, 53, 256–275. [Google Scholar] [CrossRef] - Bagchi, S. On the topological structure and properties of multidimensional (C, R) space. Symmetry
**2020**, 12, 1542. [Google Scholar] [CrossRef] - Nanda, S. Topology for Minkowski space. J. Math. Phys.
**1971**, 12, 394–401. [Google Scholar] [CrossRef] - Bharali, G.; Biswas, I. Rigidity of holomorphic maps between fiber spaces. Int. J. Math.
**2014**, 25, 1–8. [Google Scholar] [CrossRef] [Green Version] - Church, P.T.; Timourian, J.G. Fiber bundles with singularities. J. Math. Mech.
**1968**, 18, 71–90. [Google Scholar] [CrossRef] - Marathe, K.B. Minkowski spaces as fiber bundles. In C.I.M.E. Bolzano Italy; Springer: Berlin, Germany, 1970; pp. 392–401. [Google Scholar] [CrossRef]
- Zeeman, E.C. The topology of Minkowski space. Topology
**1967**, 6, 161–170. [Google Scholar] [CrossRef] [Green Version] - Fujimoto, H. On the automorphism group of a holomorphic fiber bundle over a complex space. Nagoya Math J.
**1970**, 37, 91–106. [Google Scholar] [CrossRef] [Green Version]

Space/Fibration | Geometric Property | Topological Decomposed Subspaces | Local Compactness | Group Structure |
---|---|---|---|---|

Seifert fiber space | three-Manifold | ${S}^{1}$—fiber space | Yes | Fundamental group with unique normal subgroup |

Minkowski space | four-Manifold | 3D real, 1D real spaces | Yes | Lorentz group in fiber sub-bundle |

Complex fiber bundle | Holomorphic | nD normal complex space ($n\ge 2$) | Yes | Lie group |

(C, R) space | Quasinormed | 2D complex, 1D real spaces | Yes | Two additive group varieties in fiber space |

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Bagchi, S.
Topological Analysis of Fibrations in Multidimensional (*C*, *R*) Space. *Symmetry* **2020**, *12*, 2049.
https://doi.org/10.3390/sym12122049

**AMA Style**

Bagchi S.
Topological Analysis of Fibrations in Multidimensional (*C*, *R*) Space. *Symmetry*. 2020; 12(12):2049.
https://doi.org/10.3390/sym12122049

**Chicago/Turabian Style**

Bagchi, Susmit.
2020. "Topological Analysis of Fibrations in Multidimensional (*C*, *R*) Space" *Symmetry* 12, no. 12: 2049.
https://doi.org/10.3390/sym12122049