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In differential geometry, a complete Riemannian manifold {\displaystyle } is called a Ricci soliton if, and only if, there exists a smooth vector field V {\displaystyle V} such that
for some constant λ ∈ R {\displaystyle \lambda \in \mathbb {R} }. Here Ric {\displaystyle \operatorname {Ric} } is the Ricci curvature tensor and L {\displaystyle {\mathcal {L}}} represents the Lie derivative. If there exists a function f : M → R {\displaystyle f:M\rightarrow \mathbb {R} } such that V = ∇ f {\displaystyle V=\nabla f} we call {\displaystyle } a gradient Ricci soliton and the soliton equation becomes
Note that when V = 0 {\displaystyle V=0} or f = 0 {\displaystyle f=0} the above equations reduce to the Einstein equation. For this reason Ricci solitons are a generalization of Einstein manifolds.