On Sequential Warped Products Whose Manifold Admits Gradient Schouten Harmonic Solitons
Abstract
:1. Introduction and Motivations
- Static spacetimes are characterized by a timelike Killing vector field that is orthogonal to a family of hypersurfaces. Sequential static spacetimes involve a series of such static slices.
- Analyzing these spacetimes using gradient Schouten harmonic solitons could reveal stability properties and geometric structures that might otherwise be hidden. Specifically, the flow governed by these solitons could smooth out irregularities in the spacetime fabric, leading to more stable configurations.
- Analyzing sequential GRW spacetimes involves understanding how these spacetimes evolve or transition from one state to another.
- Studying these transitions using gradient Schouten harmonic solitons could provide a unified framework to analyze curvature and geometric flows in cosmological models, potentially offering new insights into the behavior of the universe on large scales.
- Both the sequential static and GRW spacetimes can be analyzed in a combined framework using the properties of gradient Schouten harmonic solitons. This could lead to a more comprehensive understanding of spacetime dynamics.
- The interplay between static slices and cosmological models could reveal transitional behaviors and critical points in spacetime evolution.
- Development of new mathematical tools and techniques to handle the complexities of these spacetimes.
- Potential applications in quantum gravity and string theory, where the understanding of spacetime at different scales is crucial.
- Contributions to the study of black holes, cosmological singularities, and the overall geometric structure of the universe.
2. Basic Formulas and Notations
2.1. Sequentail Warped Product Manifolds
- 1.
- .
- 2.
- 3.
- .
- 4.
- .
- 5.
- .
- 6.
- .
- 7.
- 8.
- 9.
- .
- 10.
- 11.
- 12.
- 13.
- 14.
- 15.
- 1.
- .
- 2.
- 3.
- 4.
2.2. Proof of Main Theorem
- (a)
- If , thenMoreover, is Einstein with , such that
- (b)
- If , then
- (c)
- If , thenwhere and are the gradient of the function f.
3. Applications in Sequential Standard Static Spacetime
- (a)
- If , thenand together with the following
- (b)
- If , then
- (c)
- If , then
4. Applications in Generalized Robertson–Walker Spacetime
- (a)
- If , thenand together is Einstein with , such that
- (b)
- If , then
- (c)
- If , then
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
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Yan, L.; Li, Y.; Mofarreh, F.; Ali, A.; Laurian-Ioan, P. On Sequential Warped Products Whose Manifold Admits Gradient Schouten Harmonic Solitons. Mathematics 2024, 12, 2451. https://doi.org/10.3390/math12162451
Yan L, Li Y, Mofarreh F, Ali A, Laurian-Ioan P. On Sequential Warped Products Whose Manifold Admits Gradient Schouten Harmonic Solitons. Mathematics. 2024; 12(16):2451. https://doi.org/10.3390/math12162451
Chicago/Turabian StyleYan, Lixu, Yanlin Li, Fatemah Mofarreh, Akram Ali, and Pişcoran Laurian-Ioan. 2024. "On Sequential Warped Products Whose Manifold Admits Gradient Schouten Harmonic Solitons" Mathematics 12, no. 16: 2451. https://doi.org/10.3390/math12162451
APA StyleYan, L., Li, Y., Mofarreh, F., Ali, A., & Laurian-Ioan, P. (2024). On Sequential Warped Products Whose Manifold Admits Gradient Schouten Harmonic Solitons. Mathematics, 12(16), 2451. https://doi.org/10.3390/math12162451