Nilpotency of the Ricci operator of pseudo-Riemannian solvmanifolds
2
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
A pseudo-Riemannian solvmanifold is a solvable Lie group endowed with a left invariant pseudo-Riemannian metric. In this short note, we investigate the nilpotency of the Ricci operator of pseudo-Rie\-mannian solvmanifolds. We focus on a special class of solvable Lie groups whose Lie algebras can be expressed as a one-dimensional extension of a nilpotent Lie algebra $\mathbb{R}D\ltimes \mathfrak{n}$, where $D$ is a derivation of $\mathfrak{n}$ whose restriction to the center of $\mathfrak{n}$ has at least one real eigenvalue. The main result asserts that every solvable Lie group belonging to this special class admits a left invariant pseudo-Riemannian metric with nilpotent Ricci operator. As an application, we obtain a complete classification of three-dimensional solvable Lie groups which admit a left invariant pseudo-Riemannian metric with nilpotent Ricci operator.
Abstract
A pseudo-Riemannian solvmanifold is a solvable Lie group endowed with a left invariant pseudo-Riemannian metric. In this short note, we investigate the nilpotency of the Ricci operator of pseudo-Rie\-mannian solvmanifolds. We focus on a special class of solvable Lie groups whose Lie algebras can be expressed as a one-dimensional extension of a nilpotent Lie algebra $\mathbb{R}D\ltimes \mathfrak{n}$, where $D$ is a derivation of $\mathfrak{n}$ whose restriction to the center of $\mathfrak{n}$ has at least one real eigenvalue. The main result asserts that every solvable Lie group belonging to this special class admits a left invariant pseudo-Riemannian metric with nilpotent Ricci operator. As an application, we obtain a complete classification of three-dimensional solvable Lie groups which admit a left invariant pseudo-Riemannian metric with nilpotent Ricci operator.
참고문헌 (17)
- R. M. Arroyo, "On the signature of the Ricci curvature on nilmanifolds", Transf. Groups (2022)[DOI]
- A. L. Besse, "Einstein Manifolds" (1987), Springer
- M. Boucetta, "On Einstein Lorentzian nilpotent Lie groups", J. Pure Appl. Algebra, 224(12), pp.106443~ (2020)[DOI]
- D. Conti, "Diagram involutions and homogeneous Ricciflat metrics", Manuscripta Math, 165(3-4), pp.381~413 (2021)[DOI]
- D. Conti, "Einstein nilpotent Lie groups", J. Pure Appl. Algebra, 223(3), pp.976~997 (2019)[DOI]
- D. Conti, "Ricci-flat and Einstein pseudoriemannian nilmanifolds", Complex Manifolds, 6(1), pp.170~193 (2019)[DOI]
- D. Conti, "Indefinite Einstein metrics on nice Lie groups", Forum Math, 32(6), pp.1599~1619 (2020)[DOI]
- D. Conti, "Indefinite nilsolitons and Einstein solvmanifolds", J. Geom. Anal, 32(3), pp.88~ (2022)
- W. A. de Graaf, "Classification of solvable Lie algebras", Experiment. Math, 14(1), pp.15~25 (2005)
- M. B. Djiadeu Ngaha, "The signature of the Ricci curvature of left-invariant Riemannian metrics on nilpotent Lie groups", Differential Geom. Appl, 47, pp.26~42 (2016)[DOI]
- J. Heber, "Noncompact homogeneous Einstein spaces", Invent. Math, 133(2), pp.279~352 (1998)[DOI]
- J. Lauret, "A canonical compatible metric for geometric structures on nilmanifolds", Ann. Global Anal. Geom, 30(2), pp.107~138 (2006)[DOI]
- A. Medina, "Algebres de Lie et produit scalaire invariant", Ann. Sci. Ecole Norm. Sup, 18(3), pp.553~561 (1985)
- I. D. Miatello, "Ricci curvature of left invariant metrics on solvable unimodular Lie groups", Math. Z, 180(2), pp.257~263 (1982)[DOI]
- J. Milnor, "Curvatures of left invariant metrics on Lie groups", Advances in Math, 21(3), pp.293~329 (1976)[DOI]
- K. Nomizu, "Left-invariant Lorentz metrics on Lie groups", Osaka Math. J, 16(1), pp.143~150 (1979)
- B. O’Neill, "Semi-Riemannian geometry" (1983), Academic Press, Inc