arXiv research

The Riemann curvature tensor is a central mathematical tool in Einstein's theory of general relativity. Its related eigenproblem plays an important role in mathematics and physics. We extend M-eigenvalues for the elasticity tensor to the Riemann curvature tensor. The definition of M-eigenproblem of the Riemann curvatur…

We illustrate the flow or wave character of the metrics and curvatures of evolving manifolds, introducing the Riemann flow and the Riemann wave via the bialternate product Riemannian metric. This kind of evolutions are new and very natural to understand certain flow or wave phenomena in the nature as well as the geomet…

Paper examines conditions making Riemann solitons trivial and estimates their scalar curvature.

Estimates curvature for holomorphic maps on Riemann surfaces.

New tensors reveal full curvature structure from Riemann tensor.

The paper generalizes Riemann curvature for manifolds with discontinuous metrics.

Constructs metrics on Riemann surfaces with singularities.

Local fractional derivatives affect Riemann curvature tensor to zero.

Extends mean curvature to surfaces in Riemann-Cartan geometry with torsion.

A zero mean curvature surface in the Lorentz-Minkowski 3-space is said to be of Riemann-type if it is foliated by circles and at most countably many straight lines in parallel planes. We classify all zero mean curvature surfaces of Riemann-type according to their causal characters, and as a corollary, we prove that if …

The paper studies new curvature properties in Finsler geometry.

The article investigates almost Riemann solitons and gradient almost Riemann solitons in a specific type of manifold.

This is an elementary observation that the symmetry properties of the Riemann curvature tensor can be (efficiently) expressed as SL(2)-invariance.

Survey on metrics with conic singularities on Riemann surfaces.

Generalized Gauss-Bonnet formula for conical metrics on compact Riemann surfaces.

Some examples of three-dimensional metrics of constant curvature defined by solutions of nonlinear integrable differential equations and their generalizations are constructed. The properties of Riemann extensions of the metrics of constant curvature are studied. The connection with the theory of normal Riemann spaces a…

We show that on every ${\sf RCD}$ spaces it is possible to introduce, by a distributional-like approach, a Riemann curvature tensor. Since after the works of Petrunin and Zhang-Zhu we know that finite dimensional Alexandrov spaces are ${\sf RCD}$ spaces, our construction applies in particular to the Alexandrov setting.…

We study the properties of Modified Riemann extensions evolving under Ricci flow. We obtain the necessary and sufficient condition for modified Riemann extension under Ricci flow to stay as modified Riemann extension. We also discuss the properties of the curvature tensors under Ricci flow.

In this paper we present several curvature estimates and convergence results for solutions of the Ricci flow. The curvature estimates depend on smallness of certain local space-time integrals of the norm of the Riemann curvature tensor, while the convergence results require finiteness of space-time integrals of the nor…

New symmetries found in Riemann-Cartan geometries.

Let $h$ be a complete metric of Gaussian curvature $K_0$ on a punctured Riemann surface of genus $g \geq 1$ (or the sphere with at least three punctures). Given a smooth negative function $K$ with $K=K_0$ in neighbourhoods of the punctures we prove that there exists a metric conformal to $h$ which attains this function…

Proves a conjecture about Riemann surfaces using PDEs.

In this paper the rate relations of Riemann, conformal, conharmonic and Weyl curvature tensors under Yamabe flow are studied. Modified Riemann extensions under Yamabe flow is discussed. The paper ends with remarks on some standard metrics.

The Kazdan-Warner problem is solved for Riemann surfaces with smooth boundaries.

New proof shows special surfaces have finite type.

Defines new invariants for Riemann-Finsler manifolds, generalizing Preissman's theorem.

In this article we give a complete description of the evolution of an area decreasing map $f:M\to N$ induced by its mean curvature in the situation where $M$ and $N$ are complete Riemann surfaces with bounded geometry, $M$ being compact, for which their sectional curvatures $σ_M$, $σ_N$ satisfy $\minσ_M\ge\supσ_N$.

A 3D metric conformally related to Arnold cat fast dynamo metric: ${ds_{A}}^{2}=e^{-λz}dp^{2}+e^{λz}dq^{2}+dz^{2}$ is shown to present a behaviour of non-dynamos where the magnetic field exponentially decay in time. The Riemann-Christoffel connection and Riemann curvature tensor for the Arnold and its conformal counter…

We study the canonical metric on a compact Riemann surface of genus at least two. While it is known that the canonical metric is of nonpositive curvature, we show that its Gaussian curvatures are not bounded away from zero nor negative infinity when the surface is close to the compactification divisor of Riemann's modu…

We introduce the concept of singular values for the Riemann curvature tensor, a central mathematical tool in Einstein's theory of general relativity. We study the properties related to the singular values, and investigate five typical cases to show its relationship to the Ricci scalar and other invariants.

A geometric flow based in the Riemann-Christoffel curvature tensor that in two dimensions has some common features with the usual Ricci flow is presented. For $n$ dimensional spaces this new flow takes into account all the components of the intrinsic curvature. For four dimensional Lorentzian manifolds it is found that…

Study on solitons in deformed Kenmotsu manifolds with specific vector fields.

In the literature we see that after introducing a geometric structure by imposing some restrictions on Riemann-Christoffel curvature tensor, the same type structure given by imposing same restriction on other curvature tensors being studied. The main object of the present paper is to study the equivalency of various ge…

Let $X$ be a compact connected Riemann surface of genus $g\geq 0$, and let ${\rm Sym}^d(X)$, $d \ge 1$, denote the $d$-fold symmetric product of $X$. We show that ${\rm Sym}^d(X)$ admits a Hermitian metric with negative Chern scalar curvature if and only if $g \geq 2$, and positive Chern scalar curvature if and only if…

Minimal surfaces with negative curvature found in large spheres.

We study the Weil-Petersson geometry for holomorphic families of Riemann Surfaces equipped with the unique conical metric of constant curvature -1.

We generalized Xiang, Qi and Wei's results on the M-eigenvalues of Riemann curvature tensor to higher dimensional conformal flat manifolds. The expression of M-eigenvalues and M-eigenvectors are found in our paper. As a special case, M-eigenvalues of conformal flat Einstein manifold have also been discussed, and the co…

Riemann zero mean curvature examples in the Lorentz-Minkowski space are surfaces with zero mean curvature foliated by circles contained in parallel planes. In contrast to the Euclidean case, this family of surfaces presents new and rich features because of the variety of types of circles. In this paper, we give a geome…

We approach the problem of uniformization of general Riemann surfaces through consideration of the curvature equation, and in particular the problem of constructing Poincaré metrics (i.e., complete metrics of constant negative curvature) by solving the equation $Δu - e^{2u} = K_0(z)$ on general open surfaces. A few oth…

Given the Riemann, or the Weyl, or a generalized curvature tensor K, a symmetric tensor $b_{ij}$ is named `compatible' with the curvature tensor if $b_i{}^m K_{jklm} + b_j{}^m K_{kilm} + b_k{}^m K_{ijlm} = 0$. Amongst showing known and new properties, we prove that they form a special Jordan algebra, i.e. the symmetriz…

The paper constructs Markov partitions for geodesic flow on hyperbolic surfaces.

New findings on gradient expanding Ricci solitons with finite scalar curvature ratio.

We study the spectral geometry of the Riemann curvature tensor for Pseudo-Riemannian manifolds and provide some examples illustrating the phenomena which can arise in the higher signature setting. Dedication: This paper is dedicated to the memory of our colleague Prof.G. Tsagas who studied the spectral geometry of Lapl…

Derdzinski and Shen's theorem on the restrictions posed by a Codazzi tensor on the Riemann tensor holds more generally when a Riemann-compatible tensor exists. Several properties are shown to remain valid in this broader setting. Riemann compatibility is equivalent to the Bianchi identity of the new "Codazzi deviation …

We address the problem of surface inpainting, which aims to fill in holes or missing regions on a Riemann surface based on its surface geometry. In practical situation, surfaces obtained from range scanners often have holes where the 3D models are incomplete. In order to analyze the 3D shapes effectively, restoring the…

Given an smooth function $K <0$ we prove a result by Berger, Kazhdan and others that in every conformal class there exists a metric which attains this function as its Gaussian curvature for a compact Riemann surface of genus $g>1$. We do so by minimizing an appropriate functional using elementary analysis. In particula…

Defines curvature for spectral triples and applies to θ-deformations.

Study on constant curvature immersions of surfaces into flag manifolds.