arXiv research

The paper studies Berwald scalar curvature properties in Finsler geometry.

New vanishing theorems for genera derived under almost nonnegative Ricci curvature.

The study classifies metrics with vanishing curvature on complex manifolds.

Study cylindrical symmetric Finsler metrics with vanishing Douglas curvature.

We give a complete classification of homogeneous $(α,β)$-metrics with positive flag curvature and vanishing S-curvature.

Study Bochner formula on metric measure spaces for vanishing Betti numbers.

Note refutes examples of Landsberg surfaces with vanishing flag curvature.

Study on Berwald-Weyl curvature with projective invariance and vanishing results.

Sprays with vanishing X-curvature are studied in this paper.

Vanishing theorem on CR manifolds with non-negative curvature.

Study finds first integrals in Finsler metrics with vanishing χ-curvature.

Study Ricci curvature of homogeneous Finsler spaces with specific metrics.

In this Note, we prove that every m-th root Finsler metric with isotropic Landsberg curvature reduces to a Landsberg metric. Then, we show that every m-th root metric with almost vanishing H-curvature has vanishing H-curvature.

New rigidity results for tensors on non-compact manifolds with curvature conditions.

In this paper, we consider orthogonal Ricci curvature $Ric^{\perp}$ for Kähler manifolds, which is a curvature condition closely related to Ricci curvature and holomorphic sectional curvature. We prove comparison theorems and a vanishing theorem related to these curvature conditions, and construct various examples to i…

We obtain a classification theorem for non Kaehler nearly Kaehler manifolds with vanishing Bochner curvature tensor (introduced by Tricerri and Vanhecke).

Classifies surfaces with no Gaussian curvature.

Study Finsler metrics with vanishing Landsberg curvature.

The Weil-Petersson metric for the moduli space of Riemann surfaces has negative sectional curvature. Surfaces represented in the complement of a compact set in the moduli space have short geodesics. At such surfaces the Weil-Petersson metric is approximately a product metric. An almost product metric has sections with …

New curvature conditions imply vanishing of Betti numbers for certain manifolds.

Vanishing theorem for certain tensor fields on compact Hermitian manifolds.

We study curvature properties of four-dimensional almost Hermitian manifolds with vanishing Bochner curvature tensor as defined by Tricerri and Vanhecke. We give local structure theorems for such Kaehler manifolds, and find out several examples related to the theorems.

The aim of this paper is to complete the local classification of minimal hypersurfaces with vanishing Gauss-Kronecker curvature in a 4-dimensional space form. Moreover, we give a classification of complete minimal hypersurfaces with vanishing Gauss-Kronecker curvature and scalar curvature bounded from below.

The paper proves vanishing and finiteness theorems for p-harmonic 1-forms.

We investigate 3-dimensional complete minimal hypersurfaces in the hyperbolic space $\mathbb{H}^{4}$ with Gauss-Kronecker curvature identically zero. More precisely, we give a classification of complete minimal hypersurfaces with Gauss-Kronecker curvature identically zero, nowhere vanishing second fundamental form and …

For complete Riemannian manifolds with vanishing Bach tensor and positive constant scalar curvature, we provide a rigidity theorem characterized by some pointwise inequalities. Furthermore, we prove some rigidity results under an inequality involving $L^{\frac{n}{2}}$-norm of the Weyl curvature, the traceless Ricci cur…

It is proved that if an almost Hermitian manifold of dimension greater than 4 has vanishing (classical) Bochner curvature tensor and is not Kaehlerian at a point, then it is flat in a neighbourhood of this point.

We give a necessary and sufficient condition on a Randers space for the existence of a measure for which Shen's S-curvature vanishes everywhere. Moreover, such a measure coincides with the Busemann-Hausdorff measure up to a constant multiplication.

Introduces new curvature concept for Kähler manifolds.

Characterizes two-dimensional generalized Berwald metrics with vanishing S-curvature.

We obtain a vanishing theorem for the kernel of a Dirac operator on a Clifford module twisted by a sufficiently large power of a line bundle, whose curvature is non-degenerate at any point of the base manifold. In particular, if the base manifold is almost complex, we prove a vanishing theorem for the kernel of a $\spi…

The paper examines properties of spherical Finsler metrics, proving their semi-C-reducibility and conditions for vanishing mean stretch curvature.

In this paper, we study one of the open problems in Finsler geometry which presented by Matsumoto-Shimada about the existence of P-reducible metric which is not C-reducible. For this aim, we study a class of Finsler metrics called generalized P-reducible metrics that contains the class of P-reducible metrics. We prove …

The paper extends a vanishing theorem for hypersurfaces in aspherical manifolds.

An $(α,β)$-metric is defined by a Riemannian metric $α$ and $1$-form $β$. In this paper, we study a known class of two-dimensional $(α,β)$-metrics of vanishing S-curvature. We determine the local structure of those metrics and show that those metrics are Einsteinian (equivalently, isotropic flag curvature) but generall…

We prove the non-vanishing of the CMC flux of the boundaries of certain Riemannian manifolds with constant mean curvature.

The study classifies parallel mean curvature spheres in a sphere-hyperbolic product space.

In this note, we affirm the partial answer to the long open Conjecture which states that any closed embeddable strictly pseudoconvex CR $3$-manifold admits a contact form $θ$ with the vanishing CR $Q$-curvature. More precisely, we deform the contact form according to an CR analogue of $Q$%-curvature flow in a closed st…

We extend the vanishing theorem for the Seiberg-Witten invariants of a manifold with positive scalar curvature to the case when the curvature is allowed to be negative on a set of small volume. (The precise curvature bounds are described in the paper.) The idea is to combine the method of `semigroup domination' with th…

Negative curvature manifolds have vanishing bounded volume class if and only if Cheeger constant is positive.

Let T be the standard torus of revolution in R^3 with radii b and 1, 0<b<1. Let αbe a (p,q) torus curve on T. We show that there are points of zero curvature on αfor only one value of the variable radius of T, b=p^2/(p^2+q^2). The curve αhas non-vanishing curvature for all other values of b. Moreover, for this value of…

In 2001, Zhongmin Shen asked if it is possible for two projectively related Finsler metrics to have the same Riemann curvature tensor, [14, page 184]. In this paper, we provide an answer to this question, within the class of Finsler metrics of scalar flag curvature. In Theorem 3.1, we show that the answer is negative, …

Study new warped metrics with vanishing Douglas curvature.

New Bochner technique for foliations with non-negative Ricci curvature.

Vanishing theorems show holomorphic tensor fields on certain Kähler manifolds are trivial.

We prove a vanishing and estimation theorem for the $p^{\text{th}}$-Betti number of closed $n$-dimensional Riemannian manifolds with a lower bound on the average of the lowest $n-p$ eigenvalues of the curvature operator. This generalizes results due to D. Meyer, Gallot-Meyer, and Gallot. For example, in dimensions $n=5…

The paper constructs singularities for Lagrangian flow in Gibbons-Hawking spaces with vanishing mean curvature.

Let $M^n$ be a compact K$\ddot{a}$hler manifold with almost nonnegative Ricci curvature and nonzero first Betti number. We show that the holomorphic Euler number of $M^n$ vanishes, which gives a new obstruction for compact complex manifolds admitting K$\ddot{a}$hler metrics with almost nonnegative Ricci curvature. A cr…