arXiv research

This paper finds a metric on \(S^2 imes T^2\) with strictly positive biorthogonal curvature using an affine connection with antisymmetric torsion.

The abstract discusses metrics with positive biorthogonal curvature on 5-manifolds.

We prove that S^2 x S^2 satisfies an intermediate condition between having metrics with positive Ricci and positive sectional curvature. Namely, there exist metrics for which the average of the sectional curvatures of any two planes tangent at the same point, but separated by a minimum distance in the 2-Grassmannian, i…

We classify, up to homeomorphisms, the closed simply-connected 4-manifolds that admit a Riemannian metric for which averages of pairs of sectional curvatures of orthogonal planes are positive.

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 manifolds with nonnegative biorthogonal curvature. In particular, we obtain a partial a…

Study classifies special 4D shapes with certain curvature.

In this note we classify compact 4-manifolds with harmonic Weyl tensor and nonnegative biorthogonal curvature

The goal of this paper is to investigate the rigidity of 4-dimensional manifolds involving some pinching curvature conditions. To this end, we make use of the approach of biorthogonal curvature which is weaker than the sectional curvature. Here, we prove a rigidity result for 4-dimensional compact manifolds under a sui…

In this note we prove that a four-dimensional compact oriented half-confor\-mally flat Riemannian manifold $M^4$ is topologically $\mathbb{S}^{4}$ or $\mathbb{C}\mathbb{P}^{2},$ provided that the sectional curvatures all lie in the interval $[\frac{3\sqrt{3}-5}{4},\,1].$ In addition, we use the notion of biorthogonal (…

The famous pinching problem says that on a compact simply connected $n$-manifold if its sectional curvature satisfies $K_{min} > (1/4)K_{max} > 0$, then the manifold is homeomorphic to the sphere. In [8, problem 12], S. T. Yau proposed the following problem: If we replace $K_{max}$ by the scalar curvature, can we deduc…

The goal of this article is to establish estimates involving the Yamabe minimal volume, mixed minimal volume and some topological invariants on compact 4-manifolds. In addition, we provide topological sphere theorems for compact submanifolds of spheres and Euclidean spaces, provided that the full norm of the second fun…

We discuss multiscale representations of discrete manifold-valued data. As it turns out that we cannot expect general manifold-analogues of biorthogonal wavelets to possess perfect reconstruction, we focus our attention on those constructions which are based on upscaling operators which are either interpolating or midp…

It is shown that, in four dimensions, it is possible to introduce coordinates so that an analytic metric locally takes block diagonal form. i.e. one can find coordinates such that $g_{αβ} = 0$ for $(α, β) \in S$ where $S = {(1, 3), (1, 4), (2, 3), (2, 4)}$. We call a coordinate system in which the metric takes this for…

The paper constructs non-Riemannian Einstein solutions on $S^2 imes T^2$ using cohomologically calibrated affine connections.

Extends Perelman's theorem to positive intermediate curvature conditions.

Proves metrics with positive intermediate Ricci curvature on complex manifolds.

Preserves positive intermediate curvature on manifolds.

The paper explores conditions for positive scalar curvature on manifolds with boundaries and their doubles.

We begin a systematic study of a curvature condition (strongly positive curvature) which lies strictly between positive curvature operator and positive sectional curvature, and stems from the work of Thorpe in the 1970s. We prove that this condition is preserved under Riemannian submersions and Cheeger deformations, an…

In this short note we discuss some recent results about two-positive Ricci curvature and their applications to positive Einstein curvature.

A complex disproves a curvature property.

Curvature of Bazaikin spaces fully characterized, with unique quasi-positive metric.

If $π:M\rightarrow B$ is a Riemannian Submersion and $M$ has positive sectional curvature, O'Neill's Horizontal Curvature Equation shows that $B$ must also have positive curvature. We show there are Riemannian submersions from compact manifolds with positive Ricci curvature to manifolds that have small neighborhoods of…

Sharp curvature condition implies spherical space form structure.

Study finds open manifolds without complete metrics with positive scalar curvature.

Sharp volume growth ratio for 3D manifolds with positive scalar curvature.

Proves positive Euler characteristic for certain manifolds with positive second intermediate Ricci curvature.

We introduce some new curvature quantities such as conformal Ricci curvature and bi-Ricci curvature and extend the classical Myers theorem under these new curvature conditions. Moreover, we are able to obtain the Myers type theorem for minimal submanifolds in ambient manifolds with positive bi-Ricci curvature. Some top…

The study classifies graphs on surfaces with positive curvature properties.

In this paper, we give a new generalization of positive sectional curvature called positive weighted sectional curvature. It depends on a choice of Riemannian metric and a smooth vector field. We give several simple examples of Riemannian metrics which do not have positive sectional curvature but support a vector field…

New metric with negative curvature found near positive-curvature spaces.

Study finds metrics with positive intermediate Ricci curvature on specific low-dimensional manifolds.

The study examines symmetries in spaces with positive or non-negative curvature.

The paper explores curvature positivity on Kähler and quasi-Kähler flag manifolds.

We classify homogeneous reversible Finsler metrics with positive Flag curvature. We show that if G/H admits a G invariant reversible Finsler metric with positive Flag curvature, then up to a few low dimensional spaces, it also admits a G invariant Riemannian metric with positive sectional curvature. For the exceptions,…

Study Bismut-Griffiths-positivity in non-Kähler manifolds under Hermitian curvature flows.

3-manifolds with positive scalar curvature and bounded geometry are contractible.

Surveying problems with positive curvature forms, focusing on Ricci flow.

Ricci flow preserves positive sectional curvature on homogeneous spheres

The paper proves finite Urysohn 1-width for 4 and 5 manifolds with positive biRicci curvature.

Study torsion obstructions to positive scalar curvature on manifolds.

Curvature flows in hyperbolic space preserve positive sectional curvature and contract to a point.

Abstract: Characterizes spaces with positive scalar curvature.

The paper proves uniformization for specific curvature types on manifolds.

New curvature concept shows certain manifolds can't have positive curvature.

Developing a singular dimension descent method for positive scalar curvature obstructions

New metrics found for 6k-dimensional manifolds with positive Ricci curvature.

The study finds infinitely many different geometries for odd-dimensional manifolds with positive Ricci curvature.