Inverts operators near parallel Ricci metric on non-compact manifolds.
problem Inverting curvature operators near a parallel Ricci metric.
method Local invertibility of operators in Sobolev spaces.
result Operators are locally invertible near a parallel Ricci metric.
The study identifies types of manifolds using variational formulas and integral-differential formulas.
problem Identifying specific types of manifolds based on curvature properties.
method Established variational formulas for Ricci curvature bounds and used them to identify manifolds.
result Constant curvature, Einstein, and Ricci parallel manifolds identified with specific formulas.
New metrics with special curvature properties are shown to be parallel in certain Lie groups.
problem Characterizing metrics with harmonic curvature in Lie groups.
method Analyzing left invariant metrics on solvable and low-dimensional Lie groups.
result Left invariant metrics with harmonic curvature are Ricci-parallel in solvable Lie groups and Lie groups of dimension ≤6.
The paper classifies special Riemannian manifolds with cyclic parallel Ricci tensor.
problem Classifying Riemannian manifolds with specific properties.
method Analyzing solutions to a specific partial differential equation under cyclic parallel Ricci tensor conditions.
result Classification of manifolds with positive static triples, critical metrics, and total scalar curvature.
We descrive examples of metrics in the conformal class [g] on complete conformally flat Riemannian manifolds (M,g]. These metrics have a constant scalar curvature and an harmonic curvature with non parallel Ricci tensor.
Let (M,g) be a compact riemannian manifold without boundary., with parallel Rici curvature. We show that some operators, affine relatively to the Ricci curvature,are locally invertible, near the metric g
Special Liouville metrics with Ricci-like conditions are determined by elliptic functions.
problem Characterizing Liouville metrics with Ricci-like conditions in complex space forms.
method Analyzing necessary conditions for induced metrics of parallel mean curvature surfaces and proving the existence of specific Liouville metrics.
result Explicit determination of special Liouville metrics with Ricci-like conditions by elliptic functions.
Paper classifies Einstein-type manifolds with parallel Ricci tensor.
problem Classifying Einstein-type manifolds with specific curvature properties.
method Deduced Bochner-type identity and used it to show rigidity results.
result Found conditions for classifying Einstein-type manifolds with parallel Ricci tensor.
The study classifies gradient Ricci solitons with specific vector fields.
problem Characterizing gradient Ricci solitons with closed conformal vector fields.
method Analyzing properties of gradient Ricci solitons with constant scalar curvature and closed conformal vector fields.
result Gradient Ricci solitons with these properties are isometric to specific spaces.
Study shows almost conformal isometry between Sub-Semi-Riemannian metrics and solves a Ricci equation.
problem Finding almost conformal isometry between Sub-Semi-Riemannian metrics.
method Analyzes conditions for Lp almost conformal isometry and solvability of Ricci equation.
result Shows solvability of a Ricci equation without closeness assumptions.
New criterion for Ricci-flat manifolds with non-vanishing Rosenberg index.
problem Existence of parallel spinors on Ricci-flat manifolds.
method Generalization of existing results to manifolds with non-vanishing Rosenberg index.
result Every closed connected Ricci-flat spin manifold of dimension ≥ 2 with non-vanishing Rosenberg index has special holonomy.
The paper studies f-stability of hypersurfaces in gradient Ricci solitons.
problem Estimating the f-stability index of constant weighted mean curvature hypersurfaces. method Analyzes hypersurfaces in shrinking gradient Ricci solitons with parallel fields.
result Provides an estimate for the f-stability index and necessary conditions for equality. Entropy defined for submanifolds; applies to mean curvature flow limits of surfaces.
problem Entropy for submanifolds in Riemannian manifolds.
method Entropy defined and shown to be monotone along mean curvature flow.
result Partial regularity of mean curvature flow limits of surfaces.
Study pp-waves with lightlike parallel spinors in vacuum spacetimes.
problem Characterize pp-waves with lightlike parallel spinors in vacuum spacetimes.
method Parametrize pp-wave spacetimes, show correspondence with Riemannian metrics, prove parallel spinor condition.
result A pp-wave spacetime with a lightlike parallel spinor corresponds to a Ricci-flat metric with a parallel spinor.
Stability and rigidity of Ricci-flat ALE manifolds proven.
problem Stability and rigidity of Ricci-flat ALE manifolds.
method Proved stability and rigidity of ALE manifolds with a parallel spinor under Ricci flow, given initial metrics close in Lp∩L∞. result Strong decay rates prove positive scalar curvature rigidity in Lp for each p∈[1,n−2n). A. Derdzinki [D] gave examples of Riemannian metrics with harmonic curvature and non parallel Ricci tensor on some compact manifolds (M,g] . We examine their existence as well as their number wich naturally depends on the geometry of the manifolds.
Study on Ricci-like solitons on specific geometric manifolds, finding properties and conditions.
problem Characterizing Ricci-like solitons on Sasaki-like almost contact B-metric manifolds.
method Analyzing cases with specific potential fields and studying curvature conditions.
result Found conditions for the potential to have constant length and manifold to be η-Einstein. Study classifies 3D Einstein manifolds with cyclic Ricci tensor.
problem Classifying Einstein manifolds with specific tensor properties.
method Derived integral formula involving tensor D for classification.
result Obtained rigidity results for 3D manifolds.
The paper studies curvature identities and solitons on Spin(7)-manifolds.
problem Curvature identities and solitons on Spin(7)-manifolds.
method Analyzes the curvature and torsion of Spin(7)-manifolds, proving identities and conditions.
result Conditions for closed torsion and implications for Ricci flatness and solitons.
The paper proves rigidity of submanifolds in space forms with certain curvature conditions.
problem Proving rigidity of submanifolds in space forms with specific curvature constraints.
method Analyzing the integral Ricci curvature of submanifolds with parallel mean curvature.
result Submanifolds satisfying certain conditions are shown to be totally umbilical spheres.
The paper studies parallel spinor flows on 3D Cauchy hypersurfaces and provides initial data characterizations.
problem Characterizing parallel spinors on Ricci flat Lorentzian four-manifolds.
method Evolution flow defined by parallel spinors, proving preservation of constraints, solving left-invariant flows.
result Initial data characterization of parallel spinors on Ricci flat Lorentzian four-manifolds.
Study of η-Ricci solitons on Kenmotsu 3-manifolds.
problem Exploring η-Ricci solitons on Kenmotsu 3-manifolds. method Examined various types of η-Ricci solitons on Kenmotsu 3-manifolds, including those with specific curvature conditions. result Existence of proper η-Ricci solitons on Kenmotsu 3-manifolds. Discrete forms of the scalar, sectional and Ricci curvatures are constructed on simplicial piecewise flat triangulations of smooth manifolds, depending directly on the simplicial structure and a choice of dual tessellation. This is done by integrating over volumes which include appropriate samplings of hinges for each …
Study H-type foliations in sub-Riemannian geometry.
problem Understand and classify H-type foliations in sub-Riemannian geometry.
method Introduce and study H-type foliations, prove curvature bounds, and classify foliations with parallel structures.
result Prove curvature bounds and classify H-type foliations with parallel structures.
Study submanifolds in curved spaces with specific curvature bounds.
problem Investigate submanifolds in curved spaces with Ricci pinched conditions.
method Analyze submanifolds in Riemannian space forms with a lower Ricci curvature bound.
result Eliminate the need for mean curvature vector field to be parallel.
Study Brownian motion on Perelman's almost Ricci-flat manifold, proving convergence to Ricci flow limits.
problem Characterize Brownian motion and stochastic transport on Perelman's manifold.
method Construct sequences of projected Brownian motions and stochastic parallel transports, analyze Laplace and horizontal Laplacian martingale problems.
result Convergence of projected Brownian motions and stochastic parallel transports to Ricci flow limits as No∞. I apply the algebraic classification of self-adjoint endomorphisms of R2,2 provided by their Jordan canonical form to the Ricci curvature tensor of four-dimensional neutral manifolds and relate this classification to an algebraic classification of the Ricci curvature spinor. These results parallel similar re…
The paper pinches curvature in expanding Ricci solitons.
problem Curvature pinching in expanding Ricci solitons.
method Hamilton-Ivey type curvature pinching estimates.
result Three-dimensional Hamilton-Ivey type curvature pinching theorem.
In this paper we obtain a simple upper bound for the infimum of the Ricci curvatures of a complete Riemannian manifold with nonzero injectivity radius i(M) depending only on of the i(M). In case of rigidity the Riemannian manifold must be an Euclidean sphere(Euclidean space) conform the injectivity radius be finite(inf…
Study on Kähler Finsler manifolds with curvature bounds, proving theorems.
problem Understanding Kähler Finsler manifolds with curvature constraints.
method Analyzing partial parallelism of complex structure, proving theorems.
result Generalized comparison theorem for positively curved Kähler Finsler manifolds.
The Eisenhart problem of finding parallel tensors treated already in the framework of quasi-constant curvature manifolds in \cite{x:j} is reconsidered for the symmetric case and the result is interpreted in terms of Ricci solitons. If the generator of the manifold provides a Ricci soliton then this is i) expanding on p…
In this paper we deal with quadratic metric-affine gravity, which we briefly introduce, explain and give historical and physical reasons for using this particular theory of gravity. Further, we introduce a generalisation of well known spacetimes, namely pp-waves. A classical pp-wave is a 4-dimensional Lorentzian spacet…
The paper classifies metrics on Heisenberg group's cotangent bundle.
problem Investigating moduli spaces of left invariant metrics on cotangent bundles of Heisenberg group.
method Algebraic approach combined with geometrical tools like classification of hyperbolic plane conics.
result Detailed classification of various types of metrics and their properties.
Study curvature properties of G2 connections with skew-symmetric torsion.
problem Investigate curvature identities and solitons on G2 manifolds. method Analyzes curvature identities and properties of G2 connections with skew-symmetric torsion. result Characterizes conditions for curvature to be symmetric and Ricci flat.
Study improves eigenvalue bounds on Kähler manifolds.
problem Finding lower bounds for the first eigenvalue of the p-Laplacian. method Used a decomposition of the Hessian on Kähler manifolds with positive Ricci curvature.
result Improved Lichnerowicz type lower bound for the first nontrivial eigenvalue.
Investigates curvature properties of generalized pp-wave metric.
problem Examines curvature characteristics of generalized pp-wave metric.
method Analyzes Ricci, quasi-Einstein, and pseudosymmetric properties.
result Shows various curvature properties and sufficient conditions.
The curvature properties of a specific type of 6-manifold are explored.
problem Curvature identities on almost Calabi-Yau 6-manifolds with torsion.
method Analysis of the Nijenhuis tensor, torsion connection, and Ricci solitons.
result The curvature of the torsion connection is symmetric and vanishes if the norm of the torsion or scalar curvature is constant.
In this paper we obtain a splitting theorem for the symmetric diffusion operator Δφ=Δ−⟨∇φ,∇⟩ and a non-constant C3 function f in a complete Riemannian manifold M, under the assumptions that the Ricci curvature associated with Δφ satisfies Ricφ(∇f,∇f)≥0, that $|…
Investigates curvature properties of Robinson-Trautman metric.
problem Examines curvature characteristics of Robinson-Trautman metric.
method Analyzes various pseudosymmetric structures and properties of the metric.
result The metric exhibits multiple curvature properties including Roter type, 2-quasi-Einstein, and Riemann compatible.
Study minimal graphs on non-negative Ricci curvature manifolds.
problem Minimal graphs with linear growth on manifolds with non-negative Ricci curvature.
method New gradient estimate for minimal graphs and heat equation techniques.
result Non-constant minimal graphs force tangent cones to split off a line.
Study on pseudo-Riemannian metrics on Lie groups, finding new non-Einstein examples.
problem Characterizing and finding non-Einstein pseudo-Riemannian metrics on Lie groups.
method Analyzing left invariant metrics, using double extension process, and constructing examples.
result Construction of infinitely many new explicit examples of non-Einstein pseudo-Riemannian metrics on Lie groups.
Locally homogeneous Lorentzian three-manifolds with recurrect curvature are special examples of Walker manifolds, that is, they admit a parallel null vector field. We obtain a full classification of the symmetries of these spaces, with particular regard to symmetries related to their curvature: Ricci and matter colline…
The paper studies para-Kenmotsu manifolds and their properties.
problem Characterizing and studying properties of para-Kenmotsu manifolds.
method Using tensor equations and curvature conditions to characterize and study properties of para-Kenmotsu manifolds.
result Para-Kenmotsu manifolds with certain curvature conditions are of constant negative curvature −1. We derive a bound on the L∞-norm of the covariant derivative of Laplace eigensections on general Riemannian vector bundles depending on the diameter, the dimension, the Ricci curvature of the underlying manifold, and the curvature of the Riemannian vector bundle. Our result implies that eigensections with sma…
The paper proves a gap theorem for almost non-negatively curved manifolds.
problem Proving a gap theorem for almost non-negatively curved manifolds.
method Two novel technical tools: controlling the spreading of minimal geodesics and Ricci flow smoothing.
result Closed manifolds with bounded geometry are diffeomorphic to torus bundles.
Uniqueness of asymptotic limits for Ricci-flat manifolds with linear volume growth is proven.
problem Proving uniqueness of asymptotic limits for noncollapsed Ricci flat manifolds with linear volume growth.
method Relating uniqueness to the existence of a harmonic function asymptotic to a Busemann function, proving uniqueness via a monotone quantity.
result Proves uniqueness of the asymptotic limit and establishes a polynomial convergence rate.
In this work we consider periodic spherically symmetric metrics of constant positive scalar curvature on the n-dimensional cylinder called pseudo-cylindric metrics. These metrics belong to the conformal class [g0] of the Riemannian product S1×Sn−1 : a circle of length T crossed with the (n-1)-dimension…
Study torsion and curvature in ACYT and AHKT 8-manifolds.
problem Characterize torsion and curvature in ACYT and AHKT 8-manifolds.
method Analyzes properties of Nijenhuis tensors, Ricci tensors, and torsion forms on ACYT and AHKT 8-manifolds.
result Closed torsion condition and Ricci flatness for ACYT 8-manifolds.