# On an Indefinite Metric on a Four-Dimensional Riemannian Manifold

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition 1**.

**Lemma 1**.

**Remark 1**.

## 3. Circles in the Two-Plane ${\mathbf{\beta}}_{\mathbf{1}}$

**Theorem 1**

**Case (A)**- $\phi \in \left(\frac{\pi}{4},\frac{\pi}{2}\right)$ and ${R}^{2}>0$;
**Case (B)**- $\phi \in \left(\frac{\pi}{2},\frac{3\pi}{4}\right)$ and ${R}^{2}<0$.

#### 3.1. Length of a Circle with Respect to $\tilde{g}$

**Theorem 2**.

**Proof**.

**Proposition 1**.

#### 3.2. Area of a Circle with Respect to $\tilde{g}$

**Theorem 3**.

**Proof**.

**Proposition 2**.

**Proof**.

#### 3.3. Circulation and Flux with Respect to $\tilde{g}$

**Theorem 4**.

**Proof**.

**Theorem 5**.

**Theorem 6**.

**Corollary 1**.

- (a)
- $T=-C,$ in case $\phi =\frac{\pi}{3}$;
- (b)
- $T=C,$ in case $\phi =\frac{2\pi}{3}$.

## 4. Circles in the Two-Plane ${\mathbf{\beta}}_{\mathbf{2}}$

**Lemma 2**

**Theorem 7**

**Case (A)**- $\phi \in \left(\frac{\pi}{4},\frac{\pi}{3}\right)$ and ${R}^{2}>0$;
**Case (B)**- $\phi \in \left(\frac{2\pi}{3},\frac{3\pi}{4}\right)$ and ${R}^{2}<0$.

#### 4.1. Length of a Circle with Respect to $\tilde{g}$

**Theorem 8**.

**Proof**.

**Proposition 3**.

#### 4.2. Area of a Circle with Respect to $\tilde{g}$

**Theorem 9**.

**Proof**.

**Proposition 4**.

**Proof**.

#### 4.3. Circulation and Flux with Respect to $\tilde{g}$

**Theorem 10**.

**Proof**.

**Theorem 11**.

**Proof**.

**Theorem 12**.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Dokuzova, I. Curvature properties of 4-dimensional Riemannian manifolds with a circulant structure. J. Geom.
**2017**, 108, 517–527. [Google Scholar] [CrossRef] - Dokuzova, I.; Razpopov, D. Four-dimensional almost Einstein manifolds with skew-circulant structures. J. Geom.
**2020**, 111, 9. [Google Scholar] [CrossRef] - Dzhelepov, G.; Dokuzova, I.; Razpopov, D. Spheres and circles with respect to an indefinite metric of a 4-dimensional Riemannian manifold with skew-circulant structures. arXiv
**2023**, arXiv:2301.03675. [Google Scholar] - Davis, P.J. Circulant Matrices; A Wiley-Interscience Publication. Pure and Applied Mathematics; John Wiley and Sons: New York, NY, USA, 1979. [Google Scholar]
- Garayar-Leyva, G.G.; Osman, H.; Estrada-López, J.J.; Moreira-Tamayo, O. Skew-Circulant-Matrix-Based Harmonic-Canceling Synthesizer for BIST Applications. Sensors
**2022**, 22, 2884. [Google Scholar] [CrossRef] [PubMed] - Gray, R.M. Toeplitz and circulant matrices: A Review, Found. Trends Commun. Inf. Theory
**2006**, 2, 155–239. [Google Scholar] [CrossRef] - Liu, Z.; Chen, S.; Xu, W.; Zhang, Y. The eigen-structures of real (skew) circulant matrices with some applications. Comp. Appl. Math.
**2019**, 38, 178. [Google Scholar] [CrossRef] - Ng, M.K. Circulant and skew-circulant splitting methods for Toeplitz systems. J. Computat. Appl. Math.
**2003**, 159, 101–108. [Google Scholar] [CrossRef] - Zhaolin, J.; Yao, J.; Lu, F. On Skew Circulant Type Matrices Involving Any Continuous Fibonacci Numbers. Abst. Appl. Anal.
**2014**, 2014, 483021. [Google Scholar] - Li, Y.; Abolarinwa, A.; Alkhaldi, A.; Ali, A. Some Inequalities of Hardy Type Related to Witten-Laplace Operator on Smooth Metric Measure Spaces. Mathematics
**2022**, 10, 4580. [Google Scholar] [CrossRef] - Li, Y.; Ganguly, D. Kenmotsu Metric as Conformal η-Ricci Soliton. Mediterr. J. Math.
**2023**, 20, 193. [Google Scholar] [CrossRef] - Li, Y.; Srivastava, S.K.; Mofarreh, F.; Kumar, A.; Ali, A. Ricci Soliton of CR-Warped Product Manifolds and Their Classifications. Symmetry
**2023**, 15, 976. [Google Scholar] [CrossRef] - Boothby, W.M. An Introduction to Differentiable Manifolds and Riemannian Geometry, 2nd ed.; Pure and Applied Mathematics, 120; Academic Press, Inc.: Orlando, FL, USA, 1986. [Google Scholar]
- Clark, P. Green’s Theorem. Available online: http://alpha.math.uga.edu/~pete/handouteight.pdf (accessed on 4 March 2023).
- do Carmo, M.P. Differential Forms and Applications. Integration on Manifolds; Universitext; Springer: Berlin/Heidelberg, Germany, 2012. [Google Scholar]
- Gupta, V.G.; Sharma, P. Differential forms and its application. Int. J. Math. Anal.
**2008**, 2, 1051–1060. [Google Scholar] - Parkinson, C. The Elegance of Differential Forms in Vector Calculus and Electromagnetics. Master’s Thesis, University of Chester, Chester, UK, 2014. [Google Scholar]
- Petrello, R. Stokes’ Theorem. Master’s Thesis, California State University, Long Beach, CA, USA, 1998. [Google Scholar]
- Abe, N.; Nakanishi, Y.; Yamaguchi, S. Circles and spheres in pseudo-Riemannian geometry. Aequ. Math.
**1990**, 39, 134–145. [Google Scholar] [CrossRef] - Holmes, R.D.; Thompson, A. N-dimensional area and content in Minkowski spaces. Pac. J. Math.
**1979**, 85, 77–110. [Google Scholar] [CrossRef] - Ikawa, T. On curves and submanifolds in an indefinite-Riemannian manifold. Tsukuba J. Math.
**1985**, 9, 353–371. [Google Scholar] [CrossRef] - Lopez, R. Differential geometry of curves and surfaces in Lorentz-Minkowski space. Int. Electron. J. Geom.
**2014**, 7, 44–107. [Google Scholar] [CrossRef] - Mustafaev, Z. The ratio of the length of the unit circle to the area of the unit disk in Minkowski planes. Proc. Am. Math. Soc.
**2005**, 133, 1231–1237. [Google Scholar] [CrossRef]

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Razpopov, D.; Dzhelepov, G.; Dokuzova, I.
On an Indefinite Metric on a Four-Dimensional Riemannian Manifold. *Axioms* **2023**, *12*, 432.
https://doi.org/10.3390/axioms12050432

**AMA Style**

Razpopov D, Dzhelepov G, Dokuzova I.
On an Indefinite Metric on a Four-Dimensional Riemannian Manifold. *Axioms*. 2023; 12(5):432.
https://doi.org/10.3390/axioms12050432

**Chicago/Turabian Style**

Razpopov, Dimitar, Georgi Dzhelepov, and Iva Dokuzova.
2023. "On an Indefinite Metric on a Four-Dimensional Riemannian Manifold" *Axioms* 12, no. 5: 432.
https://doi.org/10.3390/axioms12050432