On a Simple Characterization of Conformally Flat 4-Dimensional Spaces of Neutral Signature

Year 2024, Volume: 36 Issue: 3, 295 - 300, 26.09.2024

Bahar Kırık Rácz

https://doi.org/10.7240/jeps.1521289

Abstract

In this work, the full characterization of 4-dimensional conformally flat spaces of neutral signature is given by using methods based on holonomy structure. Possible holonomy types are obtained for the spaces in question and several remarks are made. Various examples are presented related to this investigation.

Keywords

Weyl conformal curvature tensor, Conformally flat space, Neutral signature, Holonomy

Supporting Institution

TÜBİTAK 3501 - Career Development Program (CAREER)

Project Number

122F478

Thanks

This work is supported by TÜBİTAK (The Scientific and Technological Research Council of Türkiye) Career Development Program (3501) under project number 122F478.

4-Boyutlu Nötr Metrik İşaretli Konformal Düz Uzayların Basit Bir Karakterizasyonu Üzerine

Abstract

Bu çalışmada, 4-boyutlu nötr metrik işaretli konformal düz uzayların tam karakterizasyonu dolanım yapısına dayalı yöntemler kullanılarak verilmiştir. Söz konusu uzaylar için muhtemel dolanım tipleri elde edilmiş ve birtakım açıklamalar yapılmıştır. Bu araştırmaya ilişkin çeşitli örnekler sunulmuştur.

Keywords

Weyl konformal eğrilik tensörü, Konformal düz uzay, Nötr metrik işaret, Dolanım

Supporting Institution

TÜBİTAK 3501 - Kariyer Geliştirme Programı

Project Number

122F478

Thanks

Bu çalışma, Türkiye Bilimsel ve Teknolojik Araştırma Kurumu (TÜBİTAK) tarafından 122F478 Numaralı proje ile desteklenmiştir. Projeye verdiği destekten ötürü TÜBİTAK’a teşekkürlerimi sunarım.

References

• Catino, G. (2016). On conformally flat manifolds with constant positive scalar curvature. Proceedings of the American Mathematical Society, 144, 2627–2634.
• Galaev, A. S. (2013). Conformally flat Lorentzian manifolds with special holonomy groups. Sbornik: Mathematics, 204(9), 1264.
• Hall, G. S., & Lonie, D. P. (2000). Holonomy groups and spacetimes. Classical and Quantum Gravity, 17(6), 1369–1382.
• Hall, G. (2012). Projective relatedness and conformal flatness. Central European Journal of Mathematics, 10(5), 1763–1770.
• Goldman, W. M. (1983). Conformally flat manifolds with nilpotent holonomy and the uniformization problem for 3-manifolds. Transactions of the American Mathematical Society, 278(2), 573–583.
• Hawking, S. W., & Ellis, G. F. R. (1973). The Large Scale Structure of Space-time, Cambridge.
• Kuiper, N. H. (1949). On conformally-flat spaces in the large. Annals of Mathematics, 50(4), 916–924.
• Kurita, M. (1955). On the holonomy group of the conformally flat Riemannian manifold. Nagoya Mathematical Journal, 9, 161–171.
• Schoen, R., & Yau, S. T. (1988). Conformally flat manifolds, Kleinian groups and scalar curvature. Inventiones mathematicae, 92(1), 47–71.
• Wang, Z., & Hall, G. (2013). Projective structure in 4-dimensional manifolds with metric signature (+,+,-,-). Journal of Geometry and Physics, 66, 37–49.
• Hall, G. S. (2004). Symmetries and Curvature Structure in General Relativity. World Scientific.
• Kobayashi, S., & Nomizu, K. (1963). Foundations of Differential Geometry. Interscience, (vol 1), New York.
• Ambrose, W., & Singer, I. M. (1953). A theorem on holonomy. Transactions of the American Mathematical Society, 75(3), 428–443.
• Hall, G., & Kırık, B. (2015). Recurrence structures in 4-dimensional manifolds with metric of signature (+,+,-,-). Journal of Geometry and Physics, 98, 262–274.
• Hall, G. (2018). Curvature and holonomy in 4-dimensional manifolds admitting a metric. Balkan Journal of Geometry and Its Applications, 23(1), 44–57.
• Schell, J. F. (1961). Classification of four-dimensional Riemannian spaces. Journal of Mathematical Physics, 2, 202–206.
• Hall, G., & Wang, Z. (2012). Projective structure in 4-dimensional manifolds with positive definite metrics. Journal of Geometry and Physics, 62, 449–463.