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Homogeneous Ricci solitons. (English) Zbl 1315.53046
A shrinking or steady homogeneous Ricci soliton is a compact Einstein manifold or the product of a compact Einstein manifold and a Euclidean space. In the present paper, expanding homogeneous Ricci solitons are studied. The known examples of these are all isometric to algebraic Ricci soliton metrics on solvable Lie groups, called solvsolitons. Algebraic solitons on Lie groups are characterized by the existence of a derivation \(D\in{\mathrm{Der}}({\mathfrak{g}})\) such that \[ {\mathrm{Ric}} = c\,{\mathrm{Id}} + D, \] where \({\mathrm{Ric}}\) is the tensor of the type \((1,1)\).
In the paper, it is shown that any solvmanifold which is a Ricci soliton is isometric to a solvsoliton. Moreover, unless \(M\) is flat, it is simply connected and diffeomorphic to \({\mathbb{R}}^n\). This result generalizes special cases known from the literature. In particular, as a new result, it follows that an Einstein solvmanifold is simply connected.
New structure results on homogeneous Ricci solitons are obtained, with the implication that homogeneous Ricci solitons admitting transitive semi-simple isometry groups are Einstein and a new proof that compact homogeneous Ricci solitons are Einstein.
Further, a characterization of solvable Lie groups which admit soliton metrics is given.

MSC:
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
53C30 Differential geometry of homogeneous manifolds
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