arXiv research

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

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.

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…

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 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…

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 (…

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.

New method learns nonlinear projections for reduced-order modeling of complex dynamical systems.

We study different notions of Riemannian curvatures: The $p$-curvatures which interpolate between the scalar curvature and the sectional curvature, the Gauss-Bonnet-Weyl curvatures form another interpolation from the scalar curvature to the Gauss-Bonnet integrand. We bring out the $(p,q)$-curvatures, which incorporate …

Paper establishes a relation between Berwald scalar curvature and S-curvature.

New scalar curvature defined from Ollivier-Ricci curvature for graphs.

We show any Riemannian curvature model can be geometrically realized by a manifold with constant scalar curvature. We also show that any pseudo-Hermitian curvature model, para-Hermitian curvature model, hyper-pseudo-Hermitian curvature model, or hyper-para-Hermitian curvature model can be realized by a manifold with co…

Study examines preservation of curvature-adaptedness during mean curvature flow.

Paper explores entropic curvature in Markov chains, comparing it to other curvatures.

The paper studies Berwald scalar curvature properties in Finsler geometry.

The paper studies Finsler manifolds with a new curvature concept.

New proof shows holomorphic sectional curvature fully determines curvature tensor.

Given a compact four dimensional smooth Riemannian manifold $(M,g)$ with smooth boundary, we consider the evolution equation by $Q$-curvature in the interior keeping the $T$-curvature and the mean curvature to be zero and the evolution equation by $T$-curvature at the boundary with the condition that the $Q$-curvature …

The curvature-dimension condition implies a new weighted scalar curvature.

Compact shrinkers with curvature pinching conditions proven.

Study geodesic curvature of logarithmic spirals on curved surfaces.

Study on singularity behavior of mean curvature flow with bounded curvature and index.

Study on curvature in finitely generated groups, showing positive curvature in specific cases.

Introduces new curvature concept for Kähler manifolds.

The paper studies Kropina metrics with a specific curvature property.

We introduce a natural extension of the metric tensor and the Hodge star operator to the algebra of double forms to study some aspects of the structure of this algebra. These properties are then used to study new Riemannian curvature invariants, called the $(p,q)$-curvatures. They are a generalization of the $p$-curvat…

The paper examines geometric properties of a unique spacetime model.

Quantitative estimate for curvature in mean curvature flow.

New insights into SGD and generalization via shift-curvature and bias-curvature mechanisms.

Lower bounds on curvature integral for manifolds with curvature constraints.

Solves curvature problems on manifolds with negative curvature.

The paper finds and analyzes the Funk-Finsler structure in constant curvature spaces.

The flag curvature is a natural extension of the sectional curvature in Riemannian geometry, and the S-curvature is a non-Riemannian quantity which vanishes for Riemannian metrics. There are (incomplete) non-Riemannian Finsler metrics on an open subset in R^n with negative flag curvature and constant S-curvature. In th…

Paper estimates curvature of convex hypersurfaces with prescribed curvature.

A complete surface of constant mean curvature 1 (CMC-1) in hyperbolic 3-space with constant curvature -1 has two natural notions of "total curvature"-- one is the total absolute curvature which is the integral over the surface of the absolute value of the Gaussian curvature, and the other is the dual total absolute cur…

Curvature estimates prove existence of smooth hypersurfaces in hyperbolic space.

Study shows no large mean curvature fill-ins for nonnegative scalar curvature.

The abstract finds conditions for creating curves of constant curvature.

New manifolds with negative curvature limit to one with negative curvature.

Estimates mean curvature, scalar curvature, shape operator in warped products.

Study proves surfaces with constant curvature are simple shapes.