Suppose M is a compact n-dimensional manifold, n ≥ 2, with a metric gij(x, t) that evolves by the Ricci flow ∂tgij = −2Rij in M × (0, T ). We will give a simple proof of a recent result of Perelman on the non-existence of shrinking breather without using the logarithmic Sobolev inequality. It is known that Ricci flow is a very powerful tool in understanding the geometry and structure of manifol...

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 study some properties of a half-lightlike submanifoldM , of a semi-Riemannianmanifold, whose shape operator is conformal to the shape operator of its screen distribution. We show that any screen distribution S(TM) of M is integrable and the geometry of M has a close relation with the nondegenerate geometry of a leaf of S(TM). We prove some results on symmetric induced Ricci tensor and null s...

ABSTRACT. In this article, we continue the discussion of Fang-Wu (2015) to estimate the spectral gap of the Ornstein-Uhlenbeck operator on path space over a Riemannian manifold of pinched Ricci curvature. Along with explicit estimates we study the short-time asymptotics of the spectral gap. The results are then extended to the path space of Riemannian manifolds evolving under a geometric flow. ...

Walker manifolds of signature (2, 2) have been used by many authors to provide examples of Osserman and of conformal Osserman manifolds of signature (2, 2). We study questions of geodesic completeness and Ricci blowup in this context.

In this paper, we extend the general maximum principle in [NT3] to the time dependent Lichnerowicz heat equation on symmetric tensors coupled with the Ricci flow on complete Riemannian manifolds. As an application we exhibit complete Riemannian manifolds with bounded nonnegative sectional curvature of dimension greater than three such that the Ricci flow does not preserve the nonnegativity of t...

We study the ∗-Ricci operator on Hopf real hypersurfaces in complex quadric. prove that for quadric, tensor is symmetric if and only unit normal vector field singular. In following, we obtain of quadric symmetric, then both Reeb-flow-invariant Reeb-parallel. As correspondence to semi-symmetric Ricci tensor, give a classification with tensor.

These notes provides some details on the lectures 2,3,4 on the Ricci flow with surgery. They are not complete and probably contains some inaccuracies. Interested readers can find most exhaustives explanations on the Perelman's papers in [KL]. The aim of these lecture is to give the classification and the description of 3-dimensional κ-solutions. Let κ > 0 and (M n , g(t)) a solution of the Ricc...