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# Ricci Solitons and Geometry of Four-dimensional Non-reductive Homogeneous Spaces

Published:2011-12-23
Printed: Aug 2012
• Giovanni Calvaruso,
Dipartimento di Matematica "E. De Giorgi", Università del Salento, Prov. Lecce-Arnesano, 73100 Lecce, Italy
• Anna Fino,
Dipartimento di Matematica, Università di Torino, Via Carlo Alberto 10, 10123 Torino, Italy
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## Abstract

We study the geometry of non-reductive $4$-dimensional homogeneous spaces. In particular, after describing their Levi-Civita connection and curvature properties, we classify homogeneous Ricci solitons on these spaces, proving the existence of shrinking, expanding and steady examples. For all the non-trivial examples we find, the Ricci operator is diagonalizable.
 Keywords: non-reductive homogeneous spaces, pseudo-Riemannian metrics, Ricci solitons, Einstein-like metrics
 MSC Classifications: 53C21 - Methods of Riemannian geometry, including PDE methods; curvature restrictions [See also 58J60] 53C50 - Lorentz manifolds, manifolds with indefinite metrics 53C25 - Special Riemannian manifolds (Einstein, Sasakian, etc.)