In this work we construct and analyze exact solutions describing Ricci flows and nonholonomic deformations of four dimensional (4D) Taub-NUT spacetimes. It is outlined a new geometric techniques of constructing Ricci flow solutions. Some conceptual issues on spacetimes provided with generic off–diagonal metrics and associated nonlinear connection structures are analyzed. The limit from gravity/...

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...

In this work we construct and analyze exact solutions describing Ricci flows and nonholonomic deformations of four dimensional (4D) Taub-NUT spacetimes. It is outlined a new geometric techniques of constructing Ricci flow solutions. Some conceptual issues on spacetimes provided with generic off–diagonal metrics and associated nonlinear connection structures are analyzed. The limit from gravity/...

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...

One of the classical problems in differential geometry is the investigation of closed manifolds which admit Riemannian metrics with given lower bounds for the sectional or the Ricci curvature and the study of relations between the existence of such metrics and the topology and geometry of the underlying manifold. Despite many efforts during the past decades, this problem is still far from being...

1. The main result and some consequences. In 1956 E. Calabi [6] attacked the classification problem of compact euclidean space forms by means of a special construction, called the Calabi construction (see Wolf [14, p. 124]). Here we announce that the construction can be extended to compact riemannian manifolds whose Ricci curvature tensor is zero (Ricci flat). Of course, it is not known if ther...

We prove that any gradient shrinking Ricci soliton has at most Euclidean volume growth. This improves a recent result of H.-D. Cao and D. Zhou by removing a condition on the growth of scalar curvature. A complete Riemannian manifold M of dimension n is called gradient shrinking Ricci soliton if there exists f ∈ C (M) and a constant ρ > 0 such that Rij +∇i∇jf = ρgij , where Rij denotes the Ricci...

We study complete noncompact long time solutions (M, g(t)) to the Kähler-Ricci flow with uniformly bounded nonnegative holomorphic bisectional curvature. We will show that when the Ricci curvature is positive and uniformly pinched, i.e. Ri̄ ≥ cRgi̄ at (p, t) for all t for some c > 0, then there always exists a local gradient Kähler-Ricci soliton limit around p after possibly rescaling g(t) alon...

In [LY] a differential Harnack inequality was proved for solutions to the heat equation on a Riemannian manifold. Inspired by this result, Hamilton first proved trace and matrix Harnack inequalities for the Ricci flow on compact surfaces [H0] and then vastly generalized his own result to all higher dimensions for complete solutions of the Ricci flow with nonnegative curvature operator [ H2]. So...

The Ricci flow is an evolution system on metrics. For a given metric as initial data, its local existence and uniqueness on compact manifolds was first established by Hamilton [8]. Later on, De Turck [4] gave a simplified proof. In the later of 80's, Shi [20] generalized the local existence result to complete noncompact manifolds. However, the uniqueness of the solutions to the Ricci flow on co...