相关论文: Metrics of positive Ricci curvature on vector bund…

In this paper, we introduce the first Aeppli-Chern class for complex manifolds and show that the $(1,1)$- component of the curvature $2$-form of the Levi-Civita connection on the anti-canonical line bundle represents this class. We…

In this note we extend a recent result of S. Brendle [3] to Riemannian manifolds with densities and nonnegative Bakry-\'Emery Ricci curvature.

In a previous paper, we constructed complete manifolds of positive Ricci curvature with quadratically asymptotically nonnegative curvature and infinite topological type but dimension $\ge 6$. The purpose of the present paper is to use a…

A gap in the proof of the main result in reference [1] in our original submission propagated into the constructions presented in the first version of our manuscript. In this version we give an alternative proof for the existence of…

We give the first example of an open manifold with positive Ricci curvature and a non-proper Busemann function at a point. This provides counterexamples to a longtime well-known open question whether the Busemann function at a point of an…

We give a sufficient condition to rule out complete Riemannian metrics with nonnegative scalar curvature on the interiors of handlebodies. In higher dimensions, we give examples of ends of manifolds with positive scalar curvature metrics.

We provide new examples of manifolds which admit a Riemannian metric with sectional curvature nonnegative, and strictly positive at one point. Our examples include the unit tangent bundles of $CP^n$, $HP^n$ and $CaP^2$, and a family of lens…

We study closed, simply connected manifolds with positive $2^\mathrm{nd}$-intermediate Ricci curvature and large symmetry rank. In odd dimensions, we show that they are spheres. In even dimensions other than $6$, we show that they must have…

We give a proof of the fact that the upper and the lower sectional curvature bounds of a complete manifold vary at a bounded rate under the Ricci flow.

The purpose of this paper is to derive volume and other geometric information for three-dimensional complete manifolds with positive scalar curvature. In the case that the Ricci curvature is nonnegative, it is shown that the volume of the…

In this paper, we bend a closed Riemannian manifold in the conformal class, through solving a fully nonlinear equation. As a result, we prove that each metric of quasi-negative Ricci curvature is conformal to a metric with negative Ricci…

Considering pseudo-Riemannian $g$-natural metrics on tangent bundles, we prove that the condition of being Ricci soliton is hereditary in the sense that a Ricci soliton structure on the tangent bundle gives rise to a Ricci soliton structure…

We record two remarks. First, for a compact K\"ahler manifold with semi-positive holomorphic sectional curvature, the rational dimension of the MRC fibration is exactly the number of non-truly-flat directions. Second, for compact K\"ahler…

We prove a result on equivariant deformations of flat bundles, and as a corollary, we obtain two ``splitting in a finite cover'' theorems for isometric group actions on Riemannian manifolds with infinite fundamental groups, where the…

Positive $k^{\rm th}$-intermediate Ricci curvature on a Riemannian $n$-manifold, to be denoted by $\mathrm{Ric}_k > 0$, is a condition that interpolates between positive sectional and positive Ricci curvature (when $k =1$ and $k=n-1$…

We extend a positive Ricci curvature gluing theorem of Perelman to a range of positive intermediate curvature conditions, ranging from positive scalar curvature up to (and including) positive sectional curvature. As an application of this,…

The prescribed Ricci curvature problem consists in finding a Riemannian metric $g$ on a manifold $M$ such that the Ricci curvature of $g$ equals a given $(0,2)$-tensor field $T$. We survey the recent progress on this problem in the case…

We discuss in which sense general metric measure spaces possess a first order differential structure. Building on this, we then see that on spaces with Ricci curvature bounded from below a second order calculus can be developed, permitting…

Let $M$ be a Milnor sphere or, more generally, the total space of a linear $S^3$-bundle over $S^4$ with $H^4(M;\mathbb{Q})=0$. We show that the moduli space of metrics of nonnegative sectional curvature on $M$ has infinitely many path…

In this paper, we study the topology of complete noncompact Riemannian manifolds with asymptotically nonnegative Ricci curvature. We show that a complete noncompact manifold with asymptoticaly nonnegative Ricci curvature and sectional…