Surveying recent work on Kähler metrics and algebraic variety stability.
problem Understanding canonical Kähler metrics on algebraic varieties.
method Analyzing recent developments in algebraic geometry.
result Relation between canonical Kähler metrics and stability in algebraic geometry.
Study locally conformally balanced metrics on specific Lie algebras.
problem Characterize and classify locally conformally balanced metrics on almost abelian Lie algebras.
method Characterizations and classifications based on specific properties of Lie algebras.
result Classification of six-dimensional almost abelian Lie algebras with locally conformally balanced metrics.
The paper describes metrics on left Leibniz algebras, linking them to quadratic Lie algebras.
problem Understanding metrics on left Leibniz algebras and their connections to quadratic Lie algebras.
method Analyzing left multiplications, right multiplications, and bilinear forms on left Leibniz algebras.
result Left Leibniz algebras with associative metrics can be derived from their underlying quadratic Lie algebras.
The paper studies curvatures in metric Jordan algebras, proving unique connections and curvature formulas.
problem Investigating curvatures in metric Jordan algebras.
method Defined the Jordan-Levi-Civita connection, introduced curvature tensors, and proved curvature formulas.
result Every formally real Jordan algebra admits a metric of non-positive Jordan curvature and a Jordan-Einstein metric of negative Jordan scalar curvature.
An indecomposable Lie group with Riemannian bi-invariant metric is always simple and hence Einstein. For indefinite metrics this is no longer true, not even for simple Lie groups. We study the question of whether a semi-Riemannian bi-invariant metric is conformal to an Einstein metric. We obtain results for all three c…
We investigate a certain class of solvable metric Lie algebras. For this purpose a theory of twofold extensions associated to an orthogonal representation of an abelian Lie algebra is developed. Among other things, we obtain a classification scheme for indecomposable metric Lie algebras with maximal isotropic centre an…
Study on uniqueness of ad-invariant metrics in Lie algebras.
problem Uniqueness of ad-invariant metrics in Lie algebras up to automorphisms.
method Analysis of Lie algebras, cotangent Lie algebras, and specific conditions for uniqueness.
result Uniqueness of ad-invariant metric on T∗g implies solvability of g, but not conversely. Study on Hermitian metrics on Lie algebras with specific ideals.
problem Classifying Hermitian metrics on Lie algebras with abelian ideals.
method Examined unimodular Lie algebras with abelian ideals of codimension two, classified metrics.
result Classification of Bismut Kähler-like and Bismut torsion-parallel metrics.
The present paper contains a systematic study of the structure of metric Lie algebras, i.e., finite-dimensional real Lie algebras equipped with a non-degenerate invariant symmetric bilinear form. We show that any metric Lie algebra without simple ideals has the structure of a so called balanced quadratic extension of a…
The abstract discusses how Kähler-Einstein metrics relate to algebraic geometry.
problem Understanding the connection between Kähler-Einstein metrics and algebraic geometry.
method Exploring metric limits and rescalings of Kähler-Einstein metrics in relation to moduli spaces and singularities.
result Proposes tentative conjectural pictures connecting Kähler-Einstein metrics and algebraic geometry.
Characterizes non-degenerate cyclic metric Lie algebras.
problem Understanding the structure of non-solvable cyclic metric Lie algebras.
method Using sufficient conditions, cyclic quadruples, and double extension method.
result Complete characterization of non-degenerate cyclic metric Lie algebras.
The notion of Poisson manifold with compatible pseudo-metric was introduced by the author in [1]. In this paper, we introduce a new class of Lie algebras which we call a pseudo-Rieamannian Lie algebras. The two notions are strongly related: we prove that a linear Poisson structure on the dual of a Lie algebra has a com…
New examples of Lie algebras with ad-invariant metrics found.
problem Finding ad-invariant metrics on nonnice nilpotent Lie algebras.
method Introducing single extension method to construct Lie algebras with ad-invariant metrics.
result Explicit examples of nonnice nilpotent Lie algebras with ad-invariant metrics for dimensions > 10 and steps > 2.
New Einstein metrics found on specific Lie algebras.
problem Finding special Einstein metrics on solvable Lie algebras.
method Concrete procedure to construct Einstein pseudo-Kähler and para-Kähler metrics.
result Existence of Einstein pseudo-Riemannian metrics on specific Lie algebras.
In this paper we get a necessary and sufficient condition for the Ricci operator of a solvable metric Lie algebra to have at least two negative eigenvalues. In particular, this condition implies that the Ricci operator of every non-unimodular solvable metric Lie algebra or every non-abelian nilpotent metric Lie algebra…
Born Lie algebras classified up to 6D, with integrable metrics studied.
problem Classifying and understanding Born Lie algebras.
method Bicross product construction from pseudo-Riemannian Lie algebras.
result Classification of Lie algebras up to 6D with integrable Born structures.
In math.DG/0312243 we developed a general classification scheme for metric Lie algebras, i.e. for finite-dimensional Lie algebras equipped with a non-degenerate invariant inner product. Here we determine all nilpotent Lie algebras l with dim l'=2 which are used in this scheme. Furthermore, we classify all nilpotent met…
Classifies 4D metric Lie algebras with parallel skew-symmetric tensors.
problem Classifying 4D metric Lie algebras with parallel skew-symmetric tensors.
method Complete classification up to isometric isomorphism and scaling.
result Classification of 4D metric Lie algebras with parallel skew-symmetric tensors.
The paper generalizes the number of complex structures on metric Lie algebras.
problem How many orthogonal bi-invariant complex structures exist on metric Lie algebras?
method Developed a unique orthogonal decomposition into irreducible factors for metric Lie algebras.
result There are either 0 or 2^k such complex structures, with k the number of irreducible factors.
Characterizes Lie groups with specific structures and finds a correspondence between carrollian and galilean Lie algebras.
problem Understanding Lie groups with specific structures.
method Using structure theory of metric Lie algebras and defining new Lie algebras with skew-symmetric derivations.
result A canonical correspondence between carrollian and galilean Lie algebras mediated by bargmannian Lie algebras.
Study invariant CKY 2-forms on 5D Lie groups, classifying and determining their properties.
problem Classifying and understanding CKY 2-forms on 5D Lie groups.
method Classification and analysis of 5D metric Lie algebras with CKY tensors.
result First examples of CKY 2-forms on metric Lie algebras without Sasakian structures.
All candidates to the weakly-irreducible not irreducible holonomy algebras of Lorentzian manifolds are known. In the present paper metrics that realize all these candidates as holonomy algebras are given. This completes the classification of the Lorentzian holonomy algebras. Also new examples of metrics with the holono…
Every metric symplectic Lie algebra has the structure of a quadratic extension. We give a standard model and describe the equivalence classes on the level of corresponding quadratic cohomology sets. Finally, we give a scheme to classify the isomorphism classes of metric symplectic Lie algebras and give a complete list …
In this paper, we investigate the relationship between algebraic soliton metrics and soliton metrics for geometric evolution equations on Lie groups. After discussing the general relationship between algebraic soliton metrics and soliton metrics, we investigate the cross curvature flow and the second order renormalizat…
Eisenhart's theorem extended to sub-Riemannian metrics on specific Lie algebras.
problem Extending Eisenhart's theorem to sub-Riemannian metrics on step 2 distributions.
method Introducing ad-surjective step 2 nilpotent Lie algebras and extending Eisenhart's theorem.
result The theorem holds for sub-Riemannian metrics on ad-surjective step 2 distributions.
Study balanced Hermitian structures on almost abelian Lie algebras, classifying six-dimensional cases.
problem Classify balanced Hermitian structures on almost abelian Lie algebras.
method Classify six-dimensional almost abelian Lie algebras with balanced structures, investigate flow of balanced metrics and anomaly flow.
result Prove conjecture for compact almost abelian solvmanifolds with left-invariant complex structures.
A Riemannian Einstein solvmanifold is called standard, if the orthogonal complement to the nilradical of its Lie algebra is abelian. No examples of nonstandard solvmanifolds are known. We show that the standardness of an Einstein metric solvable Lie algebra is completely detected by its nilradical and prove that many c…
The projective algebra p(M;F) (i.e the collection of all projective vector fields)of a Finsler space (M;F) is a finite-dimensional Lie algebra with respect to the usual Lie bracket. The projective algebra of Einstein metrics has been perpetually studied from physical and geometrical approaches. Here, the projective alg…
Four dimensional simply connected Lie groups admitting a pseudo Kähler metric are determined. The corresponding Lie algebras are modelized and the compatible pairs (J,ω) are parametrized up to complex isomorphism (where J is a complex structure and ω is a symplectic structure). Such structure gives rise to a pseu…
In this work we investigate solvable and nilpotent Lie groups with special metrics. The metrics of interest are left-invariant Einstein and algebraic Ricci soliton metrics. Our main result shows that the existence of a such a metric is intrinsic to the underlying Lie algebra. More precisely, we show how one may determi…
The paper constructs Einstein Sasaki metrics on solvable Lie groups.
problem Constructing left-invariant Einstein pseudo-Riemannian Sasaki metrics on solvable Lie groups.
method Characterizing pseudo-Kähler structures and derivations giving rise to Sasaki-Einstein metrics.
result Classification of z-standard Sasaki solvable Lie algebras of dimension ≤7. Defines semi-symmetric metric connections on differential forms.
problem Analyzing connections on differential forms.
method Defined and studied semi-symmetric metric connections, computed their curvature and Ricci tensors, and analyzed Lie derivatives.
result Derived Gauss-Codazzi-Ricci equations and properties of canonical, Schouten, and Vrancreanu connections.
Study on pseudo-Hermitian quadratic nilpotent Lie algebras with methods and classifications.
problem Characterizing and classifying pseudo-Hermitian quadratic nilpotent Lie algebras.
method Construction methods and double extension by planes.
result Complete classification of nilpotent quadratic Lie algebras and pseudo-Hermitian metrics up to dimension 8.
Study left-invariant pseudo-Riemannian metrics on Lie groups focusing on null cone Lie algebras.
problem Characterize left-invariant pseudo-Riemannian metrics on Lie groups in the null cone.
method Use bracket flow on Lie algebra to study metrics on Lie groups.
result Classify all cases of null cone Lie algebras in signatures (1,q) and (2,q).
New concept of metric Lie algebras helps classify Lie groups.
problem Classifying Lie groups based on conformal Killing symmetric tensors.
method Introducing metric Lie algebras of Killing type and proving conditions for these algebras.
result Conditions for Lie algebras to be of Killing type with respect to any positive definite metric.
New calculus framework for vector bundles with metrics.
problem Developing calculus for vector bundles with fiber metrics.
method Adapting differential calculus to graded commutative algebras and focusing on diole and triole algebras.
result Triole algebra provides a suitable environment for vector bundle calculus with fiber metrics.
Study on nilpotent Lie algebras with specific metrics.
problem Classifying nilsolitons in nilpotent Lie algebras.
method Classification up to dimension 9, reduction to linear and polynomial equations.
result Complete classification of nice nilsolitons in various dimensions and signatures.
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.
A metric Lie algebra g is a Lie algebra equipped with an inner product. A subalgebra h of a metric Lie algebra g is said to be totally geodesic if the Lie subgroup corresponding to h is a totally geodesic submanifold relative to the left-invariant Riemannian metric defined by the inner product, on the simply connected …
Compactifies metrics on K3 surfaces with algebraic description.
problem Classify Gromov-Hausdorff limits of K3 surfaces with fixed structures or polarizations.
method Algebraic description of Gromov-Hausdorff compactification.
result Classification of Gromov-Hausdorff limits of K3 surfaces.
We give a global picture of the Ricci flow on the space of three-dimensional, unimodular, nonabelian metric Lie algebras considered up to isometry and scaling. The Ricci flow is viewed as a two-dimensional dynamical system for the evolution of structure constants of the metric Lie algebra with respect to an evolving or…
The collection of all projective vector fields on a Finsler space (M,F) is a finite-dimensional Lie algebra with respect to the usual Lie bracket, called the projective algebra denoted by p(M,F) and is the Lie algebra of the projective group P(M,F). The projective algebra p(M,F=α+β) of a Randers space is chara…
New methods find Ricci-flat metrics on specific Lie groups.
problem Finding Ricci-flat metrics on Lie groups.
method Two constructions based on gradings and filtrations.
result Every nilpotent Lie algebra of dimensions up to 9 admits an indefinite Ricci-flat metric.
In this paper we prove that a Finsler metrics has constant flag curvature if and only if the curvature of the induced nonlinear connection satisfies an algebraic identity with respect to some arbitrary second rank tensors. Such algebraic identity appears as an obstruction to the formal integrability of some operators i…
Study on flat metrics on 3D and 4D manifolds, focusing on topology and algebra.
problem Topology and algebraic structure of flat metrics on manifolds.
method Algebraic and topological descriptions of moduli spaces.
result Algebraic description and topology of moduli spaces for 4D manifolds with a single holonomy generator.
Study bubbling Kahler metrics using algebraic geometry.
problem Analyzing the degeneration of Kahler metrics with Euclidean volume growth.
method Algebraic construction of birational modifications to simplify degenerations, comparing with analytic constructions.
result Provide a framework to compare algebraic and analytic approaches to bubbling phenomena.
Extends differential calculus to triole algebras.
problem No specific problem stated; focuses on extending differential calculus.
method Generalizes diolic differential calculus to triole algebras with fiber metrics.
result Established a conceptual framework for calculus on bundles with vector-valued fiber metrics.
Classifies geodesic vectors in low-dimensional Lie algebras.
problem Stability of geodesic vectors in Lie algebras.
method Complete classification of Lyapunov stable and unstable geodesic vectors.
result Classification for metric Lie algebras of dimension 3 and 4.