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Ricci solitons and geometry of four-dimensional non-reductive homogeneous spaces. (English) Zbl 1252.53056

A Ricci soliton is a pseudo-Riemannian manifold \((M,g)\) admitting a smooth vector field \(V\) such that \[ {\mathcal{L}}_Vg+\varrho=\lambda g, \] where \({\mathcal{L}}_V\) denotes the Lie derivative in the direction of \(V\), \(\varrho\) is the Ricci tensor, and \(\lambda\) is a real number. A Ricci soliton is said to be shrinking, steady or expanding according to whether \(\lambda>0\), \(\lambda=0\) or \(\lambda<0\), respectively. Ricci solitons are the self-similar solutions of the Ricci flow and are important in understanding its singularities.
This paper is devoted to the geometry of non-reductive four-dimensional homogeneous spaces. After describing their Levi-Civita connection and curvature properties, the authors classify Einstein-like metrics and homogeneous Ricci solitons on these spaces, proving the existence of shrinking, expanding, and steady examples. Moreover, for all the non-trivial examples given, the Ricci operator is diagonalizable. Finally, the authors study invariant symplectic and complex structures on four-dimensional, non-reductive homogeneous, pseudo-Riemannian manifolds.

MSC:

53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
53C30 Differential geometry of homogeneous manifolds
58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
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