arXiv research

The paper extends Bonnet-Myers theorem for manifolds with nonnegative Ricci curvature.

The extrinsic Bonnet-Myers theorem is proven for positive Ricci curvature manifolds.

We give a complementary generalization of the extensions of Bonnet-Myers theorem obtained by Calabi and also Cheeger-Gromov-Taylor.

In this paper, we prove the extensions of Bonnet--Myers' type theorems obtained by Calabi and Cheeger--Gromov--Taylor via Bakry--Emery Ricci curvature, which generalize the results of \cite{FG, Lim1, Wan, Wang, WW, Wu}.

We prove the Myers-Steenrod theorem for local topological groups of isometries acting on pointed $\mathcal{C}^{k,α}$-Riemannian manifolds, with $k+α>0$. As an application, we infer a new regularity result for a certain class of locally homogeneous Riemannian metrics.

The Bakry-Émery-Ricci tensor is extended and comparison theorems are proven.

In this paper, we prove some compactness theorems of Myers, Ambrose, and Galloway for complete Riemannian manifold in the concept of $h$-almost Ricci tensors and generalized quasi-Einstein tensors. Also, we extend the previous theorems when $h$ has at most linear growth in the distance function.

Maximal diameter theorem for graphs with positive Ricci curvature.

The paper calculates graph Ricci curvature and finds properties of specific graph types.

Study on Kähler Finsler manifolds with curvature bounds, proving theorems.

In this paper,we prove the following Myers-type theorem: if $(M^n,g)$, $n\geq 3$, is an n-dimensional complete locally conformally flat Riemannian manifold with bounded Ricci curvature satisfying the Ricci pinching condition $Rc\geq εRg>0$, where $ε>0$ is an uniform constant, then $M^n$ must be compact.

The study proves a new positive energy theorem for manifolds with specific curvature properties.

In this paper we prove a new Myers' type diameter estimate on a complete connected Reimannian manifold which admits a bounded vector field such that the Bakry-Émery Ricci tensor has a positive lower bound. The result is sharper than previous Myers' type results. The proof uses the generalized mean curvature comparison …

The paper proves new comparison theorems for sub-Laplacian in foliations with minimal leaves.

In this paper we study the behavior of solutions of a second order differential equation. The existence of a zero and its localization allow us to get some compactness results. In particular we obtain a Myers' type theorem even in the presence of an amount of negative curvature. The technique we use also applies to the…

We prove a Bonnet-Myers type theorem for quaternionic contact manifolds of dimension bigger than 7. If the manifold is complete with respect to the natural sub-Riemannian distance and satisfies a natural Ricci-type bound expressed in terms of derivatives up to the third order of the fundamental tensors, then the manifo…

New Alexandrov-Patchwork construction for Lorentzian spaces with curvature bounds.

It is shown that if the Kato constant of the negative part of the Ricci curvature below a positive level is small, then the volume of the corresponding manifold can be bounded above in terms of the Kato constant and the total Ricci curvature. Together with the results from [5] and [6], this yields a generalization of t…

In this paper, we first prove the $f$-mean curvature comparison in a smooth metric measure space when the Bakry-Emery Ricci tensor is bounded from below and $|f|$ is bounded. Based on this, we define a Myers-type compactness theorem by generalizing the results of Cheeger, Gromov, and Taylor and of Wan for the Bakry-Eme…

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…

New proofs confirm travel time data determine simple metrics on a disc.

Sharp spectral theorems and isoperimetric inequalities for manifolds with nonnegative Ricci curvature.

Study diameter bounds on Kähler and quaternionic Kähler manifolds with positive curvature.

In this paper, we study the integral curvatures of Finsler manifolds and prove several Myers type theorems.

Let $M$ be a compact Riemannian manifold and $h$ a smooth function on $M$. Let $ρ^h(x)=\inf_{|v|=1}\left(Ric_x(v,v)-2Hess(h)_x(v,v) \right)$. Here $Ric_x$ denotes the Ricci curvature at $x$ and $Hess(h)$ is the Hessian of $h$. Then $M$ has finite fundamental group if $Δ^h-ρ^h<0$. Here $Δ^h=: Δ+2L_{\nabla h}$ is the Bis…

We prove a version of Myers-Steenrod's theorem for Finsler manifolds under minimal regularity hypothesis. In particular we show that an isometry between $C^{k,α}$-smooth (or partially smooth) Finsler metrics, with $k+α>0$, $k\in \mathbb{N} \cup \{0\}$, and $0 \leq α\leq 1$ is necessary a diffeomorphism of class $C^{k+1…

Synthetic proof shows globally hyperbolic Lorentzian spaces with specific curvature are warped products.

The paper proves Laplacian comparison theorems for modified m-Bakry-Emery Ricci tensors on Riemannian manifolds.

Paper develops formulas and theorems in Hermitian geometry.

Unified LLY Ricci curvature defined for hypergraphs.

Using an analogue of Myers' theorem for minimal surfaces and three dimensional topology, we prove the diameter sphere theorem for Ricci curvature in dimension three and a corresponding eigenvalue pinching theorem. This settles these two open problems for closed 3 manifolds with positive Ricci curvature since they are b…

Paper proves Horowitz-Myers conjecture in 3-7 dimensions.

The study establishes inequalities on Finsler manifolds with weighted Ricci curvature.

The standard Bonnet-Myers theorem says that if the Ricci scalar of a Riemannian manifold is bounded below by a positive number, then the manifold is compact. Moreover, a bound of its diameter is pointed out. The theorem was extended to Finsler manifolds. In this paper we prove that if a certain condition on the average…

On H-type sub-Riemannian manifolds we establish sub-Hessian and sub-Laplacian comparison theorems which are uniform for a family of approximating Riemannian metrics converging to the sub-Riemannian one. We also prove a sharp sub-Riemannian Bonnet-Myers theorem that extends to this general setting results previously pro…

Let $L=Δ-\nablaφ\cdot \nabla$ be a symmetric diffusion operator with an invariant measure $μ({\rm} d x)=e^{-φ(x)}{\mathfrak m}({\rm d} x)$ on a complete non-compact smooth Riemannian manifold $(M,g)$ with its volume element ${\mathfrak m}={\rm vol}_g$, and $φ\in C^2(M)$ a potential function. In this paper, we prove a L…

We explore the consequences of curvature and torsion on the topology of quaternionic contact manifolds with integrable vertical distribution. We prove a general Myers theorem and establish a Cartan-Hadamard result for almost qc-Einstein manifolds.

We obtain sharp quantitative Laplacian upper and lower estimates under no assumption on curvatures. As a result, we derive quantitative Laplacian, area and volume comparison theorems for tubes in Riemannian and Kähler manifolds under weak integral curvature assumptions. We also give some applications, such as a general…

The study bounds the effective diameter of graphs with positive Ollivier curvature.

For Riemannian manifolds with a smooth measure $(M, g, e^{-f}dv_{g})$, we prove a generalized Myers compactness theorem when Bakry--Emery Ricci tensor is bounded from below and $f$ is bounded.

Constructs metrics with negative constant scalar curvature.

We introduce a metric notion of Ricci curvature for $PL$ manifolds and study its convergence properties. We also prove a fitting version of the Bonnet-Myers Theorem, for surfaces as well as for a large class of higher dimensional manifolds.

Proves a sharp inequality for toroidal surfaces in Horowitz-Myers geon.

We introduce the notion of Bonnet-Myers and Lichnerowicz sharpness in the Ollivier Ricci curvature sense. Our main result is a classification of all self-centered Bonnet-Myers sharp graphs (hypercubes, cocktail party graphs, even-dimensional demi-cubes, Johnson graphs $J(2n,n)$, the Gosset graph and suitable Cartesian …

We develop the theory of weighted Ricci curvature in a weighted Lorentz-Finsler framework and extend the classical singularity theorems of general relativity. In order to reach this result, we generalize the Jacobi, Riccati and Raychaudhuri equations to weighted Finsler spacetimes and study their implications for the e…

This is the second article of a series or two, proving a generalisation of the uniqueness theorem of the Schwarzschild solution. The theorem to be shown classifies all (metrically complete) solutions of the static vacuum Einstein equations with compact but non-necessarily connected horizon without any further assumptio…

In this paper, we study the theory of geodesics with respect to the Tanaka-Webster connection in a pseudo-Hermitian manifold, aiming to generalize some comparison results in Riemannian geometry to the case of pseudo-Hermitian geometry. Some Hopf-Rinow type, Cartan-Hadamard type and Bonnet-Myers type results are establi…

Let the Ricci curvature of a compact Riemannian manifold be greater, at every point, than the Lie derivative of the metric with respect to some fixed smooth vector field. It is shown that the fundamental group then has only finitely many conjugacy classes. This applies, in particular, to all compact shrinking Ricci sol…