Related papers: Bunching for relatively pinched metrics

In this paper, we proved a compactness result about Riemannian manifolds with an arbitrary pointwisely pinched Ricci curvature tensor.

We show that a compact Riemannian manifold with weakly 1/4-pinched sectional curvatures is either locally symmetric or diffeomorphic to a space form.

The goal of this article is to study the pinching problem proposed by S.-T. Yau in 1990 replacing sectional curvature by one weaker condition on biorthogonal curvature. Moreover, we classify 4-dimensional compact oriented Riemannian…

In this note we consider versions of both Ricci and sectional curvature pinching for Riemannian manifold with density. In the Ricci curvature case the main result implies a diameter estimate that is new even for compact shrinking Ricci…

Exploiting the deformation method introduced by Aubin in his seminal work to construct constant negative scalar curvature metrics, we show the existence, on every closed manifold of dimension four, of a metric whose Bach tensor is pinched…

Motivated by a previous work of Zheng and the second named author, we study pinching constants of compact K\"ahler manifolds with positive holomorphic sectional curvature. In particular we prove a gap theorem following the work of Petersen…

This work proves certain general orbifold compactness results for spaces of Riemannian metrics, generalizing earlier results along these lines for Einstein metrics or metrics with bounded Ricci curvature. This is then applied to prove such…

There is a conjecture that a complete Riemannian 3-manifold with bounded sectional curvature, and pointwise pinched nonnegative Ricci curvature, must be flat or compact. We show that this is true when the negative part (if any) of the…

We consider a complete noncompact Riemannian manifold M and give conditions on a compact submanifold K of M so that the outward normal exponential map off of the boundary of K is a diffeomorphism onto M\K. We use this to compactify M and…

The class of the Riemannian almost product manifolds with nonintegrable structure is considered. Some identities for curvature tensor as certain invariant tensors and quantities are obtained.

In a complete simply connected Riemannian manifold X of pinched negative curvature, we give a sharp criterion for a subset C to be the epsilon-neighbourhood of some convex subset of X, in terms of the extrinsic curvatures of the boundary of…

We continue the study of the geometry and topology of compact submanifolds of arbitrary codimension in space forms that satisfy a pinching condition involving the length of the second fundamental form and the mean curvature. Our primary…

In this paper, we study closed four-dimensional manifolds. In particular, we show that under various new pinching curvature conditions (for example, the sectional curvature is no more than 5/6 of the smallest Ricci eigenvalue) then the…

We study the rigidity of compact submanifolds of Riemannian manifolds of arbitrary codimension that satisfy a sharp pinching condition involving the norm of the second fundamental form and the mean curvature. Without assuming that the…

In this paper we study sectional curvature bounds for Riemannian manifolds with density from the perspective of a weighted torsion free connection introduced recently by the last two authors. We develop two new tools for studying weighted…

Rank-one symmetric spaces carry a solvable group model which have a generalization to a larger class of Lie groups that are one-dimensional extensions of nilpotent groups. By examining some metric properties of these symmetric spaces, we…

We show that a compact oriented riemannian four-manifold with harmonic and pinched self-dual Weyl curvature is anti-self-dual if the type is nonpositive. The main part is to show that there is an almost-K\"ahler structure outside the zero…

In this paper we construct an almost negatively $1/4$-pinched Riemannian metric on a class of compact manifolds recently discovered by Stover and Toledo in [17]. It is known that these manifolds are K\"{a}hler and not locally symmetric.…

We consider the blowup of a point of a compact K\"ahler manifold and a metric of the form $\mu^*h + t b$ on it, where $h$ is a K\"ahler metric on the original manifold and $b$ is Hermitian form that looks like the Fubini--Study metric near…

We consider an asymptotically flat Riemannian spin manifold of positive scalar curvature. An inequality is derived which bounds the Riemann tensor in terms of the total mass and quantifies in which sense curvature must become small when the…