arXiv research

The paper proves Weyl projective rigidity for sub-Riemannian metrics and shows genericity of such metrics.

Study proves rigidity of certain gradient steady Ricci solitons with harmonic Weyl curvature.

The paper proves a Bonnesen-type inequality for the real projective plane.

Characterizes projective special complex manifolds using c-projective structures.

For the Bach-flat closed manifold with positive scalar curvature, we prove a rigidity result under a given inequality involving the Weyl curvature and the traceless Ricci curvature. Moveover, under an inequality involving $L^{\frac{n}{2}}$-norm of the Weyl curvature, the traceless Ricci curvature and the Yamabe invaria…

The abstract discusses transformations on statistical and semi-Weyl manifolds with torsion.

In this paper, we discuss the Weyl problem in warped product space. We obtain the openness, non rigidity and some applications. These results together with the a priori estimates obtained by Lu imply some existence results. Meanwhile we reprove the infinitesimal rigidity in the space forms.

Researchers decompose curvature to confirm Hopf conjecture and prove new rigidity theorems.

We define a Weyl-type curvature tensor that provides a characterisation for Finsler metrics of constant flag curvature. When the Finsler metric reduces to a Riemannian metric, the Weyl-type curvature tensor reduces to the classic projective Weyl tensor. In the general case, the Weyl-type curvature tensor differs from t…

3D projective structures can be metrized with conformal structures.

Paper introduces new Finsler metrics preserved under projective transformations.

Rigidity theorem for special metrics on 4-manifolds.

In this paper, we prove that the static triple with half harmonic Weyl curvature and positive scalar curvature must be the standard hemisphere.

The paper studies hypersurfaces in 5D space forms with topological and rigidity results.

This paper explores new Finsler metrics and their connections to generalized Sakaguchi's Theorem.

Paper proves rigidity of Poincaré-Einstein manifolds with cylindrical conformal infinities.

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

This article examines the coincidence of the projective and conformal Weyl tensors associated to a given connection D. The connection may be a general Weyl connection associated to a conformal class of metrics [g]. The main result for n>3 is that the Weyl tensors coincide iff D is the Levi-Civita connection of an Einst…

Weyl-type theorems extended to Galilei and Carroll geometries.

We derive point-wise and integral rigidity/gap results for a closed manifold with harmonic Weyl curvature in any dimension. In particular, there is a generalization of Tachibana's theorem for non-negative curvature operator. The key ingredients are new Bochner-Weitzenböck-Lichnerowicz type formulas for the Weyl tensor,…

The paper examines Sasaki-Ricci solitons and their transverse rigidity properties.

Non-compact convex sets in hyperbolic 3-space are rigid under isometries.

The study shows rigidity of Kähler-Ricci solitons on a specific 4D manifold.

We prove rigidity theorems for shrinking gradient Ricci solitons supporting the Heisenberg-Pauli-Weyl uncertainty principle with the sharp constant in $\mathbb{R}^n$. In addtion, we partially give analogous rigidity results of the Caffarelli-Kohn-Nirenberg inequalities on shrinking Ricci solitons.

The paper proves rigidity results for manifolds with special holonomy.

This paper derives new identities for the Weyl tensor on a gradient Ricci soliton, particularly in dimension four. First, we prove a Bochner-Weitzenböck type formula for the norm of the self-dual Weyl tensor and discuss its applications, including connections between geometry and topology. In the second part, we are co…

We define a Weyl-type curvature tensor of $(1,2)$-type to provide a characterization for Finsler metrics of constant flag curvature. This Weyl-type curvature tensor is projective invariant only to projective factors that are Hamel functions. Based on this aspect we construct new families of projectively related Finsler…

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…

In this paper, we prove rigidity results on gradient shrinking Ricci solitons with weakly harmonic Weyl curvature tensors. Let $(M^n, g)$ be a compact gradient shrinking Ricci soliton satisfying ${\rm Ric}_g + Ddf = ρg$ with $ρ>0$ constant. We show that if $(M,g)$ satisfies $δ\mathcal W (\cdot, \cdot, \nabla f) = 0$, t…

Study on mode stability of gravitational instantons of type D.

We extend harmonic map techniques to the setting of more general differential equations in conformal geometry. We obtain an extension of Siu's rigidity to Kahler-Weyl geometry and apply the latter to Vaisman's conjecture. Other applications include topological obstructions to the existence of Kahler-Weyl structures. Fo…

The study proves the rigidity of certain gradient Ricci solitons with constant scalar curvature.

The aim of the present paper is to provide an intrinsic investigation of projective changes in Finlser geometry, following the pullback formalism. Various known local results are generalized and other new intrinsic results are obtained. Nontrivial characterizations of projective changes are given. The fundamental proje…

Proves compatibility of light cones and projective structures.

In this paper, we consider some rigidity results for the Einstein metrics as the critical points of some known quadratic curvature functionals on complete manifolds, characterized by some point-wise inequalities. Moreover, we also provide rigidity results by the integral inequalities involving the Weyl curvature, the t…

In this paper, we prove some rigidity results for the Einstein metrics as the critical points of a family of known quadratic curvature functionals on closed manifolds, characterized by some point-wise inequalities. Moreover, we also provide a few rigidity results that involve the Weyl curvature, the trace-less Ricci cu…

Let $(M,g)$ be a noncompact complete $n$-manifold with harmonic curvature and positive Sobolev constant. Assume that $L_2$ norms of Weyl curvature and traceless Ricci curvature are finite. We prove that $(M,g)$ is Einstein if $n \ge 5$ and $L_{n/2}$ norms of Weyl curvature and traceless Ricci curvature are small enough…

Study rigidifies geometry of electrostatic systems with specific tensor properties.

We show that on a surface locally every affine torsion-free connection is projectively equivalent to a Weyl connection. First, this is done using exterior differential system theory. Second, this is done by showing that the solutions of the relevant PDE are in one-to-one correspondence with the sections of the `twistor…

Segre quartic surfaces linked to minitwistor spaces with Einstein-Weyl structures.

Projective structures are mostly rigid at the boundary but some are not.

We study rigidity results for the Einstein metrics as the critical points of a family of known quadratic curvature functionals involving the scalar curvature, the Ricci curvature and the Riemannian curvature tensor, characterized by some pointwise inequalities involving the Weyl curvature and the traceless Ricci curvat…

This study is motivated by the researches in the field of invariants of geodesic and conformal mappings presented in (T. Y. Thomas, [22]) and (H. Weyl, [25]). The Thomas projective parameter and the Weyl projective tensor are generalized in this article. Generators for vector spaces of invariants of geometric mappings …

An affine hypersurface (AH) structure is a pair comprising a conformal structure and a projective structure such that for any torsion-free connection representing the projective structure the completely trace-free part of the covariant derivative of any metric representing the conformal structure is completely symmetri…

In this paper we show that the space of nodal rational curves, which is so called a Severi variety (of rational curves), on any non-singular projective surface is always equipped with a natural Einstein-Weyl structure, if the space is 3-dimensional. This is a generalization of the Einstein-Weyl structure on the space o…

Projective rigidity of circle packings on complex surfaces proved.

We give some rigidity theorems for an n$(\geq4)$-dimensional compact Riemannian manifold with harmonic Weyl curvature, positive scalar curvature and positive constant $σ_2$. Moreover, when $n=4,$ we prove that a 4-dimensional compact locally conformally flat Riemannian manifold with positive scalar curvature and positi…

Entropy rigidity proven for 3D and higher convex projective manifolds.