Paper finds conditions for non-Einstein relative Yamabe metrics.
problem Finding relative Yamabe metrics with positive scalar curvature.
method Sufficient condition for positive constant scalar curvature metrics on manifolds with boundary.
result Examples of non-Einstein relative Yamabe metrics with positive scalar curvature.
New 8D geometry found with non-Einstein metrics.
problem Exploring new 8D geometric structures.
method Cohomogeneity one SU(3)-manifolds and perturbed quaternion-kaehler metrics.
result Found non-Einstein metrics with closed 4-forms.
In this note we are concerned with the distribution of Einstein and non-Einstein nilradicals among all nilpotent Lie groups. A nilpotent Lie group is called an Einstein, resp. non-Einstein, nilradical if it is a nilpotent Lie group which does, resp. does not, admit a left-invariant Ricci soliton metric. Using technique…
Study shows Bergman metric is non-Einstein for certain domains.
problem Characterizing the Bergman metric of specific domains.
method Analyzing pseudoconvex domains with strongly pseudoconvex polyhedral boundaries.
result Bergman metric is not Einstein for the studied domains.
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.
Classifies non-Einstein solutions to Einstein--Maxwell equations on 4D Lie algs.
problem Finding non-Einstein solutions to Einstein--Maxwell equations on 4D Lie algebras.
method Classification of left-invariant solutions up to automorphisms.
result Classification of all left-invariant non-Einstein solutions.
Existence of non-Einstein, non-shrinking Ricci solitons on quaternionic and octonionic spaces.
problem Existence of non-Einstein, non-shrinking Ricci solitons on specific geometric spaces.
method Construction of two 3-parameter families of non-Einstein, non-shrinking Ricci solitons on Hm+1 and HPm+1\{∗}, extending to a 2-parameter family on O2. result Existence of asymptotically paraboloidal steady Ricci solitons on Hm+1 and HPm+1\{∗}, with the Jensen sphere and Bourguignon--Karcher sphere as bases. Hamiltonian stationary Lagrangian spheres in Kaehler-Einstein surfaces are minimal. We prove that in the family of non-Einstein Kaehler surfaces given by the product Σ1×Σ2 of two complete orientable Riemannian surfaces of different constant Gauss curvatures, there is only a (non minimal) Hamiltonian stationary…
The requirement that a (non-Einstein) Kähler metric in any given complex dimension m>2 be almost-everywhere conformally Einstein turns out to be much more restrictive, even locally, than in the case of complex surfaces. The local biholomorphic-isometry types of such metrics depend, for each m>2, on three real param…
Ricci solitons as critical points of quadratic curvature functionals
problem Einstein metrics and Ricci solitons as critical points of quadratic Riemannian functionals
method Study of Ricci solitons as critical points of a special quadratic curvature functional
result Ricci solitons are non-Einstein critical points of these functionals
The aim of this short note is to announce the existence of a one-parameter family of left-invariant metrics on S3 admitting WK-spinors. This family contains the two non-Einstein Sasakian metrics with WK-spinors on S3, but does not contain the standard sphere S3 with Killing spinors. Moreover, any simply-connec…
The paper is devoted to the investigation of four-dimensional Kahler manifolds admitting non-affine H-projective mappings. We find all such manifolds which are non-Einstein. In the paper also Kahler manifolds admitting infinitesimal H-projective transformations are determined. It is proved that the class of Kahler mani…
We construct infinite families of non-simply connected locally conformally flat (LCF) 4-manifolds realizing rich topological types. These manifolds have strictly negative scalar curvature and the underlying topological 4-manifolds do not admit any Einstein metrics. Such 4-manifolds are of particular interest as example…
We consider dynamical stability for a modified Ricci flow equation whose stationary solutions include Einstein and Ricci soliton metrics. Our focus is on homogeneous metrics on non-compact manifolds. Following the program of Guenther, Isenberg, and Knopf, we define a class of weighted little Hölder spaces with certain …
Classifies Kähler structures with special foliations using symplectic techniques.
problem Classifying Kähler structures with specific foliations.
method Weinstein's method of constructing symplectic bundles applied to Kähler data.
result New Kähler structures and metrics are obtained.
The term "special biconformal change" refers, basically, to the situation where a given nontrivial real-holomorphic vector field on a complex manifold is a gradient relative to two Kähler metrics, and, simultaneously, an eigenvector of one of the metrics treated, with the aid of the other, as an endomorphism of the tan…
Three distinct 3D components found in local extremal Kähler metrics.
problem Characterizing extremal Kähler metrics in one dimension.
method Defined local extremal Kähler metrics and analyzed their germs.
result Space of germs of local extremal Kähler metrics in one dimension comprises three distinct R3 components. Study on Ricci solitons on tangent and unit tangent bundles.
problem Characterizing Ricci solitons on tangent and unit tangent bundles.
method Analyzing pseudo-Riemannian g-natural metrics and their Ricci soliton properties. result Classification of conformal vector fields and existence of non-Einstein Ricci solitons.
The present paper deals with the proper existence of a generalized class of recurrent manifolds, namely, hyper-generalized recurrent manifolds. We have established the proper existence of various generalized notions of recurrent manifolds. For this purpose we have presented a metric and computed its curvature propertie…
The paper extends Einstein condition to 4-manifolds using Hodge splittings.
problem Extending Einstein condition to 4-manifolds.
method Variational characterization and Hodge splitting approach.
result Admissible (g,h) pairs are critical points of a conformally invariant functional. We construct 4-dimensional Riemannian Lie groups carrying left-invariant conformal foliations with minimal leaves of codimension 2. We show that these foliations are holomorphic with respect to an (integrable) Hermitian structure which is not K\" ahler. We then prove that the Riemannian Lie groups constructed are {\it …
We produce new non-Kähler, non-Einstein, complete expanding gradient Ricci solitons with conical asymptotics and underlying manifold of the form R2×M2×⋯×Mr, where r≥2 and Mi are arbitrary closed Einstein spaces with positive scalar curvature. We also find numerical evidence for…
The study explores special Ricci-Hessian equations on Kähler manifolds and identifies three types of solutions.
problem Exploring special Ricci-Hessian equations on Kähler manifolds.
method Using the Cartan-Kähler theorem and analyzing specific cases.
result Three types of solutions are identified for special Ricci-Hessian equations on Kähler manifolds.
The focal sets of isoparametric hypersurfaces in spheres with g = 4 are all Willmore submanifolds, being minimal but mostly non-Einstein ([TY1], [QTY]). Inspired by A.Gray's view, the present paper shows that, these focal sets are all A- manifolds but rarely Ricci parallel, except possibly for the only unclassified cas…
Warped product manifolds with p-dimensional base, p=1,2, satisfy some curvature conditions of pseudosymmetry type. These conditions are formed from the metric tensor g, the Riemann-Christoffel curvature tensor R, the Ricci tensor S and the Weyl conformal curvature C of the considered manifolds. The main result of the p…
New framework for generalized Killing spinors on manifolds.
problem Formulating generalized Killing spinor equations as polyform systems.
method Developing a new framework using polyforms and algebraic relations in the Kähler-Atiyah bundle.
result Characterization of real spinor squaring map as a real algebraic variety.
New steady Ricci solitons found in even dimensions, including a four-dimensional example.
problem Existence of non-collapsed steady Ricci solitons in even dimensions.
method Construction of a family of non-collapsed, non-Kähler, non-Einstein steady Ricci solitons on complex line bundles over Kähler-Einstein manifolds.
result Existence of new non-collapsed steady Ricci solitons in even dimensions, including a four-dimensional example.
Classifies weakly Einstein curvature tensors in 4D Euclidean space.
problem Classifying algebraic curvature tensors in 4D Euclidean space.
method Algebraic formulation and geometric interpretation of weakly Einstein manifolds.
result Complete classification of non-Einstein weakly Einstein curvature tensors in dimension four.
This thesis surveys various metrics on Riemann surface spaces.
problem Various metrics on Riemann surface spaces.
method Survey of metrics and their properties.
result Equivalence of Kähler-Einstein metric to Teichmüller metric.
Study on conditions for Randers metrics to be of constant Ricci curvature.
problem Conditions for Randers metrics to be of constant Ricci curvature.
method Analysis of sufficient and necessary conditions for Randers metrics with and without strong convexity.
result Classification of Randers metrics with ∥β∥α>1 and ∥β∥α≡1. Proves existence and uniqueness of weighted metrics for smooth spaces.
problem Existence and uniqueness of weighted metrics for smooth metric measure spaces.
method Proves existence and uniqueness using weighted ambient metrics and Poincaré metrics.
result Existence and uniqueness of weighted metrics for smooth metric measure spaces.
New Finsler metrics constructed from GDW-metrics.
problem Exploring new Finsler metrics within the GDW-metric class. method Constructing new sub-classes of GDW-metrics. result Presented illustrative examples of new Finsler metrics.
The paper introduces new Finsler metrics and associated weighted quasi-metrics.
problem Geometric properties of Finsler metrics and quasi-metric spaces.
method Construction of weighted quasi-metrics associated with Finsler metrics.
result Investigation of geometric properties of weighted quasi-metric spaces.
Two metrics on graph spaces compared to Weil-Petersson.
problem Comparing metrics on graph spaces.
method Two Riemannian metrics on a moduli space of metric graphs.
result Comparison of geometric features with Weil-Petersson metric.
We prove the equivalences of several classical complete metrics on the Teichmüller and the moduli spaces of Riemann surfaces. We use as bridge two new Kähler metrics, the Ricci metric and the perturbed Ricci metric and prove that the perturbed Ricci metric is a complete Kähler metric with bounded negative holomorphic s…
New metric defined for bounded symmetric domains.
problem Defining a new metric for bounded symmetric domains.
method Using generalized Hilbert metric and Borel embedding.
result The new metric differs from Carathéodory and Bergman metrics except for complex hyperbolic space.
Study on Finsler metrics finds no P-reducible metrics with vanishing S-curvature.
problem Existence of P-reducible metrics with vanishing S-curvature.
method Investigation of generalized P-reducible metrics and proving their reduction to Berwald or C-reducible metrics.
result No concrete P-reducible (α,β)-metric with vanishing S-curvature exists. The paper explores generalized Douglas-Weyl metrics and their properties.
problem Characterizing and understanding generalized Douglas-Weyl metrics.
method Analyzing properties of (α,β)-metrics and proving conditions for being generalized Douglas-Weyl metrics. result Generalized Douglas-Weyl metrics are Berwald metrics under certain conditions.
Introduces Finslerian convolution metrics and their properties.
problem No specific problem stated; focuses on new metric concept.
method Definition and study of Finslerian convolution metrics.
result Characterization of Finslerian convolution metrics of Riemannian, Minkowskian, and Randers types.
New Douglas metrics found in a specific Finsler class.
problem Finding new Douglas metrics in Finsler geometry.
method Defined general (α,β)-metrics and solved PDEs for vanishing Douglas curvature. result Many new Douglas metrics constructed.
New balanced metrics introduced for SPD matrices, improving metric choice.
problem Lack of principles for choosing SPD matrix metrics.
method Introducing balanced metrics that relate existing metrics.
result Two new balanced metric families introduced: mixed-power-Euclidean and mixed-power-affine.
Survey of recent metric geometry in Kähler metrics space.
problem Understanding the metric geometry of Kähler metrics space.
method Survey and highlighting of recent results.
result Highlighting of open problems in the field.
The paper defines metrics from Lie groups and conjectures they are Einstein metrics.
problem Defining metrics from Lie groups.
method Analyzing metrics from specific Lie groups like unitary, orthogonal, and symplectic groups.
result Conjectures that metrics from these Lie groups are Einstein metrics.
Study on geodesics of Finsler metrics derived from Riemannian metrics.
problem Investigating geodesics in Finsler metrics derived from Riemannian metrics.
method Proved geodesic lemma for a family of Riemannian geodesic orbit metrics, analyzed geodesic graphs.
result Derived Finsler metrics have geodesic orbit property and belong to a new class of metrics.
Paper examines conditions for singular square metrics to have constant curvature.
problem Conditions for constant curvature in singular square metrics.
method Analyzes Finsler metrics, introduces singular square metrics, provides necessary and sufficient conditions.
result Necessary and sufficient conditions for constant Ricci or flag curvature in singular square metrics.
We consider geometries on the space of Riemannian metrics conformally equivalent to the widely studied Ebin L^2 metric. Among these we characterize a distinguished metric that can be regarded as a generalization of Calabi's metric on the space of Kähler metrics to the space of Riemannian metrics, and we study its geome…
New Kähler metrics generalize Calabi's and relate to Fano manifolds.
problem Existence of extremal Kähler metrics on Fano manifolds.
method Introducing σ-extremal Kähler metrics and relating them to multiplier Hermitian-Einstein metrics. result Existence of σ-extremal Kähler metrics implies existence of multiplier Hermitian-Einstein metrics on Fano manifolds. New Finsler metrics defined by Riemannian and 1-forms are studied.
problem Characterize and study properties of (α,β,γ)-metrics. method Introduced and defined (α,β,γ)-metrics, analyzed their properties, and found conditions for local projective flatness and Douglas type. result Necessary and sufficient conditions for (α,β,γ)-metrics to be locally projectively flat and Douglas type were found.