In the context of paracontact geometry, η-Ricci solitons are considered on manifolds satisfying certain curvature conditions: R(ξ,X) · S = 0, S · R(ξ,X) = 0, W2(ξ,X) · S = 0 and S · W2(ξ,X) = 0. We prove that on a para-Kenmotsu manifold (M,φ, ξ, η, g), the existence of an η-Ricci soliton implies that (M, g) is quasi-Einstein and if the Ricci curvature satisfies R(ξ,X) · S = 0, then (M, g) is Ei...

where r = ∫ Rdμ/ ∫ dμ is the average scalar curvature (R is the scalar curvature) and Ric is the Ricci curvature tensor of h. Hamilton then spectacularly illustrated the success of this method by proving, when n = 3, that if the initial Riemannian metric has strictly positive Ricci curvature it evolves through time to a positively curved Einstein metric h∞ on M . And, because n = 3, such a Riem...

The Ricci curvature of graphs, as recently introduced by Lin, Lu, and Yau following a general concept due to Ollivier, provides a new and promising isomorphism invariant. This paper presents a simplified exposition of the concept, including the so-called logistic diagram as a computational or visualization aid. Two new infinite classes of graphs with positive Ricci curvature are identified. A l...

The Ricci curvature at an isolated singularity of an immersed hypersurface exhibits a local behaviour echoing a global property of the Ricci curvature on a complete hypersurface in euclidean space (i.e., the Efimov theorem [12], that sup Ric ≥ 0). This local behaviour takes the form of a near universal bound on the decay of the Ricci curvature at a simple singularity (eg. a cone singularity) in...

We compare two approaches to Ricci curvature on non-smooth spaces, in the case of the discrete hypercube {0, 1} . While the coarse Ricci curvature of the first author readily yields a positive value for curvature, the displacement convexity property of Lott, Sturm and the second author could not be fully implemented. Yet along the way we get new results of a combinatorial and probabilistic natu...

We try to provide a visual introduction to some objects used in Riemannian geometry: parallel transport, sectional curvature, Ricci curvature, Bianchi identities... We then explain some of the strategies used to define analogues of curvature in non-smooth or discrete spaces, beginning with Alexandrov curvature and δ-hyperbolic spaces, and insisting on various notions of generalized Ricci curvat...

We prove the uniqueness of solutions of the Ricci flow on complete noncompact manifolds with bounded curvatures using the De Turck approach. As a consequence we obtain a correct proof of the existence of solution of the Ricci harmonic flow on complete noncompact manifolds with bounded curvatures. Recently there is a lot of study on the Ricci flow on manifolds by R. Hamilton [H1–6], S.Y. Hsu [Hs...

We compare two approaches to Ricci curvature on nonsmooth spaces in the case of the discrete hypercube {0, 1}N . While the coarse Ricci curvature of the first author readily yields a positive value for curvature, the displacement convexity property of Lott, Sturm, and Villani could not be fully implemented. Yet along the way we get new results of a combinatorial and probabilistic nature, includ...

We consider dimension reduction for solutions of the Kähler-Ricci flow with nonegative bisectional curvature. When the complex dimension n = 2, we prove an optimal dimension reduction theorem for complete translating KählerRicci solitons with nonnegative bisectional curvature. We also prove a general dimension reduction theorem for complete ancient solutions of the Kähler-Ricci flow with nonneg...

We give a new estimate on the lower bound for the first Dirichlet eigenvalue for the compact manifolds with boundary and positive Ricci curvature in terms of the diameter and the lower bound of the Ricci curvature and give an affirmative answer to the conjecture of P. Li for the Dirichlet eigenvalue.