arXiv research

New dynamics derived from Lie groups using 2nd order tangent groups.

Study of Riemannian geometry on tangent Lie groups of two families.

The paper defines a differential for Carnot manifold maps and constructs a tangent groupoid.

Paper defines tangent Lie algebra for non-smooth subgroups of diffeomorphism groups.

New complex structures found on tangent bundles of Lie groups.

New structure found on Lie group tangent bundle.

Tangent Lie groups have a special Riemannian metric structure.

We compute the fundamental groups of the complements of the family of real conic-line arrangements with up to two conics which are tangent to each other at two points, with an arbitrary number of tangent lines to both conics. All the resulting groups turn out to be big.

Study on Randers metrics on tangent Lie groups and their geometric properties.

We consider trivializations of second iterated bundles of a Lie group that preserve lifted group structures. With such a trivialization, we elaborate Hamiltonian dynamics on cotangent, Lagrangian dynamics on tangent bundles and, both Hamiltonian and Lagrangian dynamics on Tulczyjew's symplectic space which is tangent o…

Study of tangent spaces in diffeological spaces under Lie group actions.

We consider sets of locally finite perimeter in Carnot groups. We show that if E is a set of locally finite perimeter in a Carnot group G, then for almost every x in G with respect to the perimeter measure of E, some tangent of E at x is a vertical halfspace. This is a partial extension of a theorem of Franchi-Serapion…

This paper is the first part of a series of three papers about the fundamental groups of conic-line arrangements consist of two tangented conics and up to two additional tangented lines. In this part, we compute the local braid monodromies and the local fundamental groups of the singularities which appear in such arran…

The paper studies special metrics on tangent Lie groups.

Study geodesic Lie groups' convergence to limits with quantitative estimates.

Study Riemann-Finsler geometry on tangent bundles of Lie groups with 2D commutator subgroup.

The paper studies geometric properties of tangent Poisson-Lie groups.

Study of Lagrangian dynamics on matched Lie groups.

Study minimal rational curves on complex manifolds with isotropic VMRT.

We show that the tangent cone at the identity is not a complete quasiconformal invariant for sub-Riemannian nilpotent groups. Namely, we show that there exists a nilpotent Lie group equipped with left invariant sub-Riemannian metric that is not locally quasiconformally equivalent to its tangent cone at the identity. In…

The paper studies Riemannian metrics on tangent Lie groups using two left-invariant metrics.

Survey on finite dimensional Lie groups over real numbers.

In this paper, we present a study on the prolongations of representations of Lie algebras. We show that a tangent bundle of a given Lie algebra attains a Lie algebra structure. Then, we prove that this tangent bundle is algebraically isomorphic to the Lie algebra of a tangent bundle of a Lie group. Using these, we defi…

As a step toward proving an index theorem for hypoelliptic operators Heisenberg manifolds, including those on CR and contact manifolds, we construct an analogue for Heisenberg manifolds of Connes' tangent groupoid of a manifold $M$. As it is well known for a Heisenberg manifold $(M,H)$ the relevant notion of tangent is…

For any principal bundle $P$, one can consider the subspace of the space of connections on its tangent bundle $TP$ given by the tangent bundle $T{\cal A}$ of the space of connections ${\cal A}$ on $P$. The tangent gauge group acts freely on $T{\cal A}$. Appropriate BRST operators are introduced for quantum field theori…

Study new metrics on tangent bundles from base manifolds.

Study shows finitely generated fundamental groups for certain nonnegative Ricci curvature manifolds.

Study characterizes $k$-rectifiable sets in homogeneous groups.

Ancient Ricci flows on non-collapsed manifolds have finite fundamental groups.

We study how the notion of tangent space can be extended from smooth manifolds to diffeological spaces, which are generalizations of smooth manifolds that include singular spaces and infinite-dimensional spaces. We focus on two definitions. The internal tangent space of a diffeological space is defined using smooth cur…

Minimal vector fields on oscillator groups studied, with specific conditions for minimality.

Study on Klt varieties with trivial canonical class, focusing on holonomy and stability.

The paper studies geodesics in Carnot groups and their properties.

The derivation $d_T$ on the exterior algebra of forms on a manifold $M$ with values in the exterior algebra of forms on the tangent bundle $TM$ is extended to multivector fields. These tangent lifts are studied with applications to the theory of Poisson structures, their symplectic foliations, canonical vector fields a…

Study on geodesics and dihedral groups in lattices.

The paper introduces Carnot coordinates for Carnot manifolds, simplifying nilpotent approximation.

This work defines a categorical notion of principal bundles.

Frölicher spaces form a cartesian closed category which contains the category of smooth manifolds as a full subcategory. Therefore, mapping groups such as C^\infty(M,G) or \Diff(M), but also projective limits of Lie groups are in a natural way objects of that category, and group operations are morphisms in the category…

In this article, we derive a topological obstruction to the removal of a isolated degenerate complex tangent to an embedding of a 3-manifold into $\mathbb{C}^3$ (without affecting the structure of the remaining complex tangents). We demonstrate how the vanishing of this obstruction is both a necessary and sufficient co…

In Heisenberg groups, rectifiability is studied for subsets using $C^{1,α}$-regular surfaces.

We equip the direct limit of tangent bundles of paracompact finite dimensional manifolds with a structure of convenient vector bundle with structural group $GL(\infty,\mathbb{R}) =\underrightarrow{\lim}GL(\mathbb{R}^{n})$.

The paper extends Cartan development to infinite dimensional Lie groups.

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

Study local structure of knot group representations into SL(n,C).

Our goal in this paper is to make an attempt to find the largest Lie algebra of vector fields on the indicatrix such that all its elements are tangent to the holonomy group of a Finsler manifold. First, we introduce the notion of the curvature algebra, generated by curvature vector fields, then we define the infinitesi…

Proves rectifiability for specific metric spaces with unique tangents.

We list all the possible fundamental groups of the complements of real conic-line arrangements with two conics which are tangent to each other at two points, with up to two additional lines. For the computations we use the topological local braid monodromies and the techniques of Moishezon-Teicher and van-Kampen. We al…

Logarithmic connections on complex manifolds with trivial tangent bundle.