if there exists a smooth function f on Mn such that the Ricci tensor Rij of the metric gij satisfies the equation Rij +∇i∇jf = ρ gij for some constant ρ. For ρ = 0 the Ricci soliton is steady, for ρ> 0 it is shrinking and for ρ < 0 expanding. The function f is called a potential function of the gradient

Abstract. Consider the unnormalized Ricci flow (gij)t = −2Rij for t ∈ [0, T), where T < ∞. Richard Hamilton showed that if the curvature operator is uniformly bounded under the flow for all times t ∈ [0, T), then the solution can be extended beyond T. We prove that if the Ricci curvature

Consider the unnormalized Ricci flow (gij)t = −2Rij for t ∈ [0, T), where T < ∞. Richard Hamilton showed that if the curvature operator is uniformly bounded under the flow for all times t ∈ [0, T) then the solution can be extended beyond T. We prove that if the Ricci curvature is uniformly

Consider the unnormalized Ricci flow (gij)t = −2Rij for t ∈ [0, T), where T < ∞. Richard Hamilton showed that if the curvature operator is uniformly bounded under the flow for all times t ∈ [0, T) then the solution can be extended beyond T. We prove that if the Ricci curvature is uniformly

In [4], R. Hamilton has proved that if a compact manifold M of dimension three admits a C ∞ Riemannian metric g0 with positive Ricci curvature, then it also admits a metric g with constant positive sectional curvature, and is thus a quotient of the sphere S³. In fact, he shows that the original

Consider {(M n, g(t)), 0 ≤ t < T < ∞} as an unnormalized Ricci flow solution: dgij = −2Rij for t ∈ [0, T). Richard Hamilton shows that if the dt curvature operator is uniformly bounded under the flow for all t ∈ [0, T) then the solution can be extended over T. Natasa Sesum proves that a

For Riemannian manifolds with a measure (M, g, e −f dvolg) we prove mean curvature and volume comparison results when the ∞-Bakry-Emery Ricci tensor is bounded from below and f is bounded or ∂rf is bounded from below, generalizing the classical ones (i.e. when f is constant). This leads

Abstract not found

Abstract. The Bakry-Émery tensor gives an analog of the Ricci tensor for a Riemannian manifold with a smooth measure. We show that some of the topological consequences of having a positive or nonnegative Ricci tensor are also valid for the Bakry-Émery tensor. We show that the Bakry-Émery tensor

Abstract. Following the approach of Bryant [9] we study the intrinsic torsion of a SU(3)-manifold deriving a number of formulae for the Ricci and the scalar curvature in terms of torsion forms. As a consequence we prove that in some special cases the Einstein condition forces the vanishing