arXiv research

The Hessian geometry is the real analogue of the Kähler one. Sasakian geometry is an odd-dimensional counterpart of the Kähler geometry. In the paper, we study the connection between projective Hessian and Sasakian manifolds analogous to the one between Hessian and Kähler manifolds. In particular, we construct a Sasaki…

The paper studies efficient Hessian fitting methods for stochastic optimization.

Develops a bialgebra theory for post-Lie algebras using geometric interpretations and bilinear forms.

Study the geometry of a Lie group using Hessian and flat affine structures.

We study the class of 3-dimensional nonlinear 2-hessian equations mentioned in the text. We perform preliminary group classification on 2-hessian equation. In fact, we find additional equivalence transformation on the space (x,y,z,u,f), with the aid of a method, then we take their projections on the space (x,y,z,f), so…

In this article we show that every geodesic is rank one and the Hessian of Busemann functions is positive definite for a harmonic Damek-Ricci space, a two step solvable Lie group with a left invariant metric. Moreover, the eigenspace of the Hessian of Busemann functions on a Hadamard manifold $(M,g)$ corresponding to e…

This paper studies moduli spaces of statistical structures on Lie groups.

In this paper, we generalize all the results obtained on para-Kähler Lie algebras in Journal of Algebra {\bf 436} (2015) 61-101 to para-Kähler Lie algebroids. In particular, we study exact para-Kähler Lie algebroids as a generalization of exact para-Kähler Lie algebras. This study leads to a natural generalization of p…

New preconditioners speed up SGD on Lie groups.

Statistical Lie algebras with constant curvature are linked to locally conformally Kähler structures.

Study classifies special Hessian rank 2 hypersurfaces in 4D space.

Constructs homogeneous Kähler structures on tangent bundles of Hessian manifolds.

This paper summarizes closed-form relations for SE(3) maps and their derivatives.

Study the Hessian geometry of an ideal gas in a centrifuge.

We show that if a compact Kaehler manifold $M$ of non-negative Ricci curvature admits closed Fedosov star product then the reduced Lie algebra of holomorphic vector fields on $M$ is reductive. This comes in pair with the obstruction previously found by La Fuente-Gravy. More generally we consider the squared norm of Cah…

We show vanishing theorems of $L^2$-cohomology groups of Kodaira-Nakano type on complete Hessian manifolds. We obtain further vanishing theorems of $L^2$-cohomology groups $L^2H^{p,q}(Ω)$ on a regular convex cone $Ω$ with the Cheng-Yau metric for $p>q$.

Locally conformally Hessian manifolds are dense in radiant ones of rank 1.

On a constraint manifold we give an explicit formula for the Hessian matrix of a cost function that involves the Hessian matrix of a prolonged function and the Hessian matrices of the constraint functions. We give an explicit formula for the case of the orthogonal group ${\bf O}(n)$ by using only Euclidean coordinates …

The paper introduces statistical and geometric structures on anti-commutable pre-Leibniz algebroids.

Let $(M,g)$ be a compact Kähler manifold and $f$ a positive smooth function such that its Hamiltonian vector field $K = J\mathrm{grad}_g f$ for the Kähler form $ω_g$ is a holomorphic Killing vector field. We say that the pair $(g,f)$ is conformally Einstein-Maxwell Kähler metric if the conformal metric $\tilde g = f^{-…

Formula for Laplace-Beltrami on orthogonal group in Euclidean coords.

This research detects and identifies human-made objects in 3D point clouds using novel methods.

Let $P \subset \R^3$ be a polyhedron. It was conjectured that if $P$ is weakly convex (i. e. its vertices lie on the boundary of a strictly convex domain) and decomposable (i. e. $P$ can be triangulated without adding new vertices), then it is infinitesimally rigid. We prove this conjecture under a weak additional assu…

With the rapid development of social media sharing, people often need to manage the growing volume of multimedia data such as large scale video classification and annotation, especially to organize those videos containing human activities. Recently, manifold regularized semi-supervised learning (SSL), which explores th…

Characterizes real left symmetric algebras with positive definite Koszul form and related Kähler-Einstein structures.

The paper proves conjectures about Minkowski norms with specific symmetry groups.

In this work we deal with coverings and actions of Lie group- groupoids being a sort of the structured Lie groupoids. Firstly, we define an action of a Lie group-groupoid on some Lie group and the smooth coverings of Lie group-groupoids. Later, we show the equivalence of the category of smooth actions of Lie group-grou…

Lie groups of automorphisms of cotangent bundles of Lie groups are completely characterized and interesting results are obtained. We give prominence to the fact that the Lie groups of automorphisms of cotangent bundles of Lie groups are super symmetric Lie groups. In the cases of orthogonal Lie lgebras, semi-simple Lie…

We describe the structure of the Lie groups endowed with a left-invariant symplectic form, called symplectic Lie groups, in terms of semi-direct products of Lie groups, symplectic reduction and principal bundles with affine fiber. This description is particularly nice if the group is Hamiltonian, that is, if the left c…

In this survey, we report on the state of the art of some of the fundamental problems in the Lie theory of Lie groups modeled on locally convex spaces, such as integrability of Lie algebras, integrability of Lie subalgebras to Lie subgroups, and integrability of Lie algebra extensions to Lie group extensions. We furthe…

In this note we construct an infinite-dimensional Lie group structure on the group of vertical bisections of a regular Lie groupoid. We then identify the Lie algebra of this group and discuss regularity properties (in the sense of Milnor) for these Lie groups. If the groupoid is locally trivial, i.e. a gauge groupoid, …

In this paper, we introduce the notion of a (regular) Hom-Lie group. We associate a Hom-Lie algebra to a Hom-Lie group and show that every regular Hom-Lie algebra is integrable. Then, we define a Hom-exponential (Hexp) map from the Hom-Lie algebra of a Hom-Lie group to the Hom-Lie group and discuss the universality of …

The purpose of this paper is to show how central extensions of (possibly infinite-dimensional) Lie algebras integrate to central extensions of étale Lie 2-groups. In finite dimensions, central extensions of Lie algebras integrate to central extensions of Lie groups, a fact which is due to the vanishing of π_2 for each …

The paper integrates Lie-Leibniz triples into Lie group-rack triples.

New metrics with special curvature properties are shown to be parallel in certain Lie groups.

We extend the Nambu bracket to 1-forms. Following the Poisson-Lie case, we define Nambu-Lie groups as Lie groups endowed with a multiplicative Nambu structure. A Lie group G with a Nambu structure P is a Nambu-Lie group iff P=0 at the unit and the Nambu bracket of left (right) invariant forms is left (right) invariant.…

Regular Lie groups are infinite dimensional Lie groups with the property that smooth curves in the Lie algebra integrate to smooth curves in the group in a smooth way (an `evolution operator' exists). Up to now all known smooth Lie groups are regular. We show in this paper that regular Lie groups allow to push surprisi…

Investigate local Lie group structure of bisections over compact manifolds

Introduces $Θ$-positivity in Lie groups, generalizing Lusztig's positivity.

A Lie group is called orthogonal if it carries a bi-invariant pseudo Riemannian metric. Oscillator Lie groups constitutes a subclass of the class of orthogonal Lie groups. In this paper, we determine the Lie bialgebra structures and the solutions of the classical Yang-Baxter equation on a generic class of oscillator Li…

The paper integrates Rota-Baxter Lie algebras into Lie group structures and geometries.

Jet spaces on Carnot groups have a canonical Lie group structure.

Defines structure constants for specific geometric structures on Lie groups.

Study describes isometry groups of specific Lie groups.

Simple construction of Lie 2-groups from loop group extensions.

This paper studies actions of solvable Lie groups on nilpotent Lie groups.

Study finds all 4D Lie groups with harmonic curvature.

Study on subgroups' evolution in 3D Lie groups using mean curvature flow.