arXiv research

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

New complex structures found on tangent bundles of Lie groups.

New structure found on Lie group tangent bundle.

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.

This paper studies the infinitesimal structure of Carnot manifolds. By a Carnot manifold we mean a manifold together with a subbundle filtration of its tangent bundle which is compatible with the Lie bracket of vector fields. We introduce a notion of differential, called Carnot differential, for Carnot manifolds maps (…

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.

Using vertical and complete lifts, any left invariant Riemannian metric on a Lie group induces a left invariant Riemannian metric on the tangent Lie group. In the present article we study the Riemannian geometry of tangent bundle of two families of Lie groups. The first one is the family of special Lie groups considere…

In this paper we introduce the notion of tangent space TG of a (not necessary smooth) subgroup G of the diffeomorphism group Diff(M) of a compact manifold M. We prove that TG is a Lie subalgebra of the Lie algebra of smooth vector fields on M. The construction can be generalized to subgroups of any (finite or infinite …

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…

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…

Given a matched pair of Lie groups, we show that the tangent bundle of the matched pair group is isomorphic to the matched pair of the tangent groups. We thus obtain the Euler-Lagrange equations on the trivialized matched pair of tangent groups, as well as the Euler-Poincaré equations on the matched pair of Lie algebra…

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 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.

Given a compact Lie group $G$ with Lie algebra $\mathfrak{g}$, we consider its tangent Lie group $TG\cong G\ltimes_{\mathrm{Ad}} \mathfrak{g}$. In this short note, we prove that $TG$ admits a left-invariant naturally reductive Riemannian metric $g$ and a metric connection with skew torsion $\nabla$ such that $(TG,g,\na…

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 characterizes $k$-rectifiable sets in homogeneous groups.

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

Let $G$ be a Lie group equipped with a left invariant Randers metric of Berward type $F$, with underlying left invariant Riemannian metric $g$. Suppose that $\widetilde{F}$ and $\widetilde{g}$ are lifted Randers and Riemannian metrics arising from $F$ and $g$ on the tangent Lie group $TG$ by vertical and complete lifts…

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.

In this paper we study lifted left invariant $(α,β)$-metrics of Douglas type on tangent Lie groups. Let $G$ be a Lie group equipped with a left invariant $(α,β)$-metric of Douglas type $F$, induced by a left invariant Riemannian metric $g$. Using vertical and complete lifts, we construct the vertical and complete lifte…

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.

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…

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

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

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…

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…

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…

Proves rectifiability for specific metric spaces with unique tangents.

This short note has been written as an Oberwolfach report for the workshop "Differentialgeometrie im Grossen". We discuss properties of metric spaces that at almost all points admit a tangent metric space. We explain why, under some mild assumptions, the tangents are almost surely subFinsler Carnot groups. We mention s…

Logarithmic connections on complex manifolds with trivial tangent bundle.

New integrators for mechanical systems on Lie groups simplify based on group properties.

Characterizes magnetic unit vector fields on Lie groups.

In this note we show that for any proper action of a Banach--Lie group $G$ on a Banach manifold $M$, the corresponding tangent maps $\g \to T_x(M)$ have closed range for each $x \in M$, i.e., the tangent spaces of the orbits are closed. As a consequence, for each free proper action on a Hilbert manifold, the quotient $…

We investigate the holonomy group of singular Kähler-Einstein metrics on klt varieties with numerically trivial canonical divisor. Finiteness of the number of connected components, a Bochner principle for holomorphic tensors, and a connection between irreducibility of holonomy representations and stability of the tange…