Ricci-Yamabe Solitons Lie grubu üzerine Sol3

Abstract

Ricci–Yamabe solitonlarını, Thurston'ın sekiz model geometrisinden biri olan üç boyutlu çözülebilir Lie grubu $\mathrm{Sol}_3$ üzerinde inceliyoruz. Kanonik sol-invariant metrikin Levi-Civita bağlantısı ve Ricci tensörünü hesapladıktan sonra, bu tür solitonların varlığı için gerekli ve yeterli koşulları türetmiş ve ilgili vektör alanlarının açık bir sınıflandırmasını vermişiz. Ayrıca $\mathrm{Sol}_3$'ın hiçbir trivial olmayan gradyanı Ricci–Yamabe solitonunu kabul etmediğini kanıtlarız ve soliton vektör alanının ikili $1$-formunun bir temas yapısını tanımladığı koşulları karakterize ederiz. Bir uygulama olarak, beş kat $(\mathrm{Sol}_3, g, X, \mu_1, \mu_2)$ abartılı bir Ricci solitonu oluşturur.

Keywords

• Lie grubu $mathrm{Sol}_3$
• Ricci--Yamabe solitonları
• Temas formu
• Gradient alanı
• Thurston geometrisi
• Levi-Civita bağlantısı.

RICCI–YAMABE SOLITONS ON THE LIE GROUP Sol3

Abstract

We study Ricci–Yamabe solitons on the three-dimensional solvable Lie group $\mathrm{Sol}_3$, one of Thurston's eight model geometries. After computing the Levi-Civita connection and the Ricci tensor of the canonical left-invariant metric, we derive necessary and sufficient conditions for the existence of such solitons and give an explicit classification of the associated vector fields. We further prove that $\mathrm{Sol}_3$ admits no non-trivial gradient Ricci–Yamabe soliton, and we characterize the conditions under which the dual $1$-form of the soliton vector field defines a contact structure. As an application, we determine when the quintuple $(\mathrm{Sol}_3, g, X, \mu_1, \mu_2)$ constitutes a hyperbolic Ricci soliton.

Keywords

• Lie group $\mathrm{Sol}_3$
• Ricci--Yamabe solitons
• Contact form
• Gradient field
• Thurston geometry
• Levi-Civita connection

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