arXiv research

Study unimodular measures on Riemannian manifold spaces.

We study the ends of a generic manifold, with respect to a unimodular measure on the space of pointed Riemannian manifolds with bounded curvatures. We apply our general result to the case of surfaces and obtain as corollaries a very precise description of generic leaves for foliations with invariant measures and of quo…

The paper defines unimodularity for coisotropic Poisson spaces and discusses invariant volume forms.

The paper classifies Sasaki and Vaisman manifolds of unimodular Lie groups.

Prove an isoperimetric inequality for compact bodies in 3D contact non-unimodular Lie groups.

Study of unimodular Sasaki and Vaisman Lie groups, determining all modifications explicitly.

Unimodularity criteria for Poisson structures on foliated manifolds are established.

Study left invariant Lorentzian metrics on 3D non-unimodular Lie groups.

This work explores maximal symmetry in unimodular solvmanifolds.

Non-unimodular foliations have specific geometric properties.

The paper classifies structures on specific Lie groups.

The paper explores the connection between Poisson-Lie structures and invariant volume forms in Hamiltonian dynamics.

We define integral odd Khovanov homology of principally unimodular bipartite graph-links.

Paper generalizes results from nilpotent Lie algebras to broader types.

Proofs centerless unimodular contact Lie algebras.

Using technique of wheeled props we establish a correspondence between the homotopy theory of unimodular Lie 1-bialgebras and the famous Batalin-Vilkovisky formalism. Solutions of the so called quantum master equation satisfying certain boundary conditions are proven to be in 1-1 correspondence with representations of …

Constructs quaternionic complexes over unimodular quaternionic manifolds.

Study on conformal vector fields on Lie groups, proving properties and providing examples.

New Einstein manifolds split into symmetric and compact parts.

Study Riemannian Q-manifolds with Killing vector fields, finding them unimodular.

We construct integral bases for the SO(3)-TQFT-modules of surfaces in genus one and two at roots of unity of prime order and show that the corresponding mapping class group representations preserve a unimodular Hermitian form over a ring of algebraic integers. For higher genus surfaces the Hermitian form sometimes must…

Unimodular classification of symmetric matrix map-germs.

We consider actions of non-compact simple Lie groups preserving an analytic rigid geometric structure of algebraic type on a compact manifold. The structure is not assumed to be unimodular, so an invariant measure may not exist. Ergodic stationary measures always exist, and when such a measure has full support, we show…

We study left-invariant locally conformally Kähler structures on Lie groups, or equivalently, on Lie algebras. We give some properties of these structures in general, and then we consider the special cases when its complex structure is bi-invariant or abelian. In the former case, we show that no such Lie algebra is uni…

In this paper, we shall use a method based on the theory of extensions of left-symmetric algebras to classify complete left-invariant affine real structures on solvable non-unimodular three-dimensional Lie groups.

We obtain an exhaustive classification of totally umbilical surfaces in unimodular and non-unimodular simply-connected 3-dimensional Lie groups endowed with arbitrary left-invariant Riemannian metrics. This completes the classification of totally umbilical surfaces in homogeneous Riemannian 3-manifolds.

We compute the hyperbolic covolume of the automorphism group of each even unimodular Lorentzian lattice. The result is obtained as a consequence of a previous work with Belolipetsky, which uses Prasad's volume to compute the volumes of the smallest hyperbolic arithmetic orbifolds.

A Levi-Malcev type decomposition for $2$-step solvable Lie algebras with a complex structure

Study of symmetries in 4D Lie groups.

Let $\mathfrak{g}$ be a $2n$-dimensional unimodular Lie algebra equipped with a Hermitian structure $(J,F)$ such that the complex structure $J$ is abelian and the fundamental form $F$ is balanced. We prove that the holonomy group of the associated Bismut connection reduces to a subgroup of $SU(n-k)$, being $2k$ the dim…

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…

Exact G2-structures found on specific Lie algebras.

A unimodular complex surface is a complex 2-manifold X endowed with a holomorphic volume form. A strictly pseudoconvex real hypersurface M in X inherits not only a CR-structure but a canonical coframing as well. In this article, this canonical coframing on M is defined, its invariants are discussed and interpreted geom…

New findings on plane waves in 3D spacetimes, showing non-unimodular elliptic plane waves are unique.

We show that every unimodular Lie algebra, of dimension at most 4, equipped with an inner product, possesses an orthonormal basis comprised of geodesic elements. On the other hand, we give an example of a solvable unimodular Lie algebra of dimension 5 that has no orthonormal geodesic basis, for any inner product.

Unified theory connects local and global geometric properties of random hyperbolic polyhedra.

We determine, for all three-dimensional non-unimodular Lie groups equipped with a Lorentzian metric, the set of homogeneous geodesics through a point. Together with the results of [C] and [CM2], this leads to the full classification of three-dimensional Lorentzian g.o. spaces and naturally reductive spaces.

Anomaly flow on Lie groups finds diverse behavior in solutions.

Study on completeness of metrics on specific Lie groups.

We show that, for any regular Poisson manifold, there is an injective natural linear map from the first leafwise cohomology space into the first Poisson cohomology space which maps the Reeb class of the symplectic foliation to the modular class of the Poisson manifold. The Riemannian interpretation of those classes wil…

Study classifies biharmonic and harmonic homomorphisms between specific Lie groups.

The two-loop renormalization group flow is studied via the induced bracket flow on 3D unimodular Lie groups. A number of steady solitons are found. Some of these steady solitons come from maximally symmetric metrics that are steady, shrinking, or expanding solitons under Ricci flow, while others are not obviously relat…

Researchers find conditions for a specific curvature on Lie groups.

Novel radar waveform design for autonomous vehicles.

Characterizes G2-structures on Lie algebras with non-trivial center.

For finite dimensional real Lie algebras, we investigate the existence of an inner product having a basis comprised of geodesic elements. We give several existence and non-existence results in certain cases: unimodular solvable Lie algebras having an abelian nilradical, algebras having an abelian derived algebra, algeb…

Note on linearizing certain Nambu structures.

In this work it is shown that a necessary condition for the completeness of the geodesics of left invariant pseudo-Riemannian metrics on Lie groups is also sufficient in the case of 3-dimensional unimodular Lie groups, and not sufficient for 3-dimensional non unimodular Lie groups. As a consequence it is possible to id…