arXiv research

The paper studies billiards in symmetric tables and finds a measure bound for maximizing orbits.

Locally maximizing orbits studied in twist maps and billiards.

We formulate and prove that there are "abundant" in nilpotent orbits in real semisimple Lie algebras, in the following sense. If S denotes the collection of hyperbolic elements corresponding the weighted Dynkin diagrams coming from nilpotent orbits, then S span the maximally expected space, namely, the (-1)-eigenspace …

The main result of this paper is, that for convex billiards in higher dimensions, in contrast with 2D case, for every point on the boundary and for every $n$ there always exist billiard trajectories developing conjugate points at the $n$-th collision with the boundary. We shall explain that this is a consequence of the…

We prove that for a compact subgroup $H$ of an almost connected locally compact Hausdorff group $G$, the following properties are mutually equivalent: (1) $H$ is a maximal compact subgroup of $G$, (2) $G/H$ is contractible, (3) $G/H$ is homeomorphic to a Euclidean space, (4) $G/H$ is an AE for paracompact spaces, (5) $…

We show that the maximal orbit dimension of a simultaneous Lie group action on n copies of a manifold does not pseudo-stabilize when n increases. We also show that if a Lie group action is (locally) effective on subsets of a manifold, then the induced Cartesian action is locally free on an open subset of a sufficiently…

Graph products inherit Morse local-to-global property from their components.

Let G_R be a Lie group acting on an oriented manifold M, and let $ω$ be an equivariantly closed form on M. If both G_R and M are compact, then the integral $\int_M ω$ is given by the fixed point integral localization formula (Theorem 7.11 in [BGV]). Unfortunately, this formula fails when the acting Lie group G_R is not…

This thesis studies the symplectic structure of holomorphic coadjoint orbits, and their projections. A holomorphic coadjoint orbit O is an elliptic coadjoint orbit which is endowed with a natural invariant Kählerian structure. These coadjoint orbits are defined for a real semi-simple connected non-compact Lie group G w…

We list up all the possible local orbit types of hyperbolic or elliptic orbits for the isotropy representations of semisimple pseudo-Riemannian symmetric spaces. It is key to give a recipe to determine the local orbit types of hyperbolic principal orbits by using three kind of restricted root systems and Satake diagram…

Higher index theorem for Dirac operators on finite-volume spaces.

Study GKM actions on special manifolds with interval orbit spaces.

Study horocycle orbits in $\mathbb{Z}$-covers of hyperbolic surfaces.

Effective estimates for lattice orbits in homogeneous spaces.

Classifies symplectic torus actions up to equivariant symplectomorphism.

The paper calculates the full asymptotics of analytic torsions for compact orbifolds.

We show that, if the family \cal{O} of orbits of all vector fields on a subcartesian space P is locally finite and each orbit in \cal{O} is locally closed, then \cal{O} defines a smooth Whitney A stratification of P. We also show that the stratification by orbit type of the space M/G of orbits of a proper action of a L…

This work is concerned with Mishchenko-Fomenko subalgebras and their restrictions to the adjoint orbits in a finite-dimensional complex semisimple Lie algebra. In this setting, it is known that each Mishchenko-Fomenko subalgebra restricts to a completely integrable system on every orbit in general position. We improve …

Study orbit spaces of equivariant ANEs for proper actions of metrizable groups.

We prove a relation between the $\bar\partial_M$ cohomology of a minimal orbit $M$ of a real form $G_0$ of a complex semisimple Lie group $G$ in a flag manifold $G/Q$ and the Dolbeault cohomology of the Matsuki dual open orbit $X$ of the complexification $K$ of a maximal compact subgroup $K_0$ of $G_0$, under the assum…

The present paper is a continuation of Le Anh Vu's ones [13], [14], [15]. Specifically, the paper is concerned with the subclass of connected and simply connected MD5-groups such that their MD5-algebras $\mathcal{G}$ have the derived ideal ${\mathcal{G}}^{1} : = [ \mathcal{G},\mathcal{G} ]\equiv$ ${\bf{R}}^{3}$. We sha…

Locally convex bialgebroids reconstruct Lie groupoids of orbits.

Extends flat submanifold properties from hyperbolic plane to symmetric spaces.

New invariant $c_{sm}$ simplifies computing geometric invariants of recursive group orbits.

Let $\mathcal{G} \rightrightarrows M$ be a source locally trivial proper Lie groupoid such that each orbit is of finite type. The orbit projection $M \to M/\mathcal{G}$ is a fibration if and only if $\mathcal{G} \rightrightarrows M$ is regular.

Study geodesics on $SL(n, \mathbb{R})$ adjoint orbits.

We investigate several situations where the local homogeneity of a geometric structure on a dense open subset of a manifold implies the local homogeneity everywhere. This results in a strengthening of the conclusions in Gromov's open-dense orbit theorem. In particular, we show that any smooth closed 3-dimensional Loren…

Maximal representations in symplectic lattices proven for most cases.

Counting geodesics on compact symmetric spaces using orbit dimensions and topological data.

The goal of this diploma thesis is to give a detailed description of Kirillov's Orbit Method for the case of compact connected Lie groups. The theory of Kirillov aims at finding all irreducible unitary representations of a given Lie group $G$. The first chapter is intended to recall some facts about Lie groups. The mos…

The abstract describes a foliation of orbits for a specific class of Lie groups.

This article investigates a few questions about orbits of local automorphisms in manifolds endowed with rigid geometric structures. We give sufficient conditions for local homogeneity in a broad class of such structures, namely Cartan geometries, extending a classical result of Singer about locally homogeneous Riemanni…

In the first part of this paper, we give a global description of simply connected maximal Lorentzian surfaces whose group of isometries is of dimension 1 (i.e. with a complete Killing field), in terms of a 1-dimensional generally non-Hausdorff manifold (the space of Killing orbits) and a smooth function defined there. …

We apply the Guillemin-Lerman-Sternberg theorem to reprove a formula of Heckman for the Duistermaat-Heckman measure associated to the coadjoint action of $T$, a maximal torus of a compact semisimple Lie group $G$, on a regular coadjoint $G$-orbit in the dual space of the Lie algebra of $G$. This formula is, in an appro…

Classifies $T^k$-manifolds with continuous sections.

Study on rolling of 2D and 3D manifolds, identifying orbit dimensions.

The main result of this article is that if a $3$-manifold $M$ supports an Anosov flow, then the number of conjugacy classes in the fundamental group of $M$ grows exponentially fast with the length of the shortest orbit representative, hereby answering a question raised by Plante and Thurston in 1972. In fact we show th…

We show that the differential structure of the orbit space of a proper action of a Lie group on a smooth manifold is continuously reflexive. This implies that the orbit space is a differentiable space in the sense of Smith, which ensures that the orbit space has an exterior algebra of differenial forms, which statisfie…

In this paper we find examples of slant surfaces in the nearly Kahler six sphere. First, we characterize two-dimensional small and great spheres which are slant. Their description is given in terms of the associative 3-form in $\Im \OO .$ Later on, we classify the slant surfaces of $S^6$ which are orbits of maximal tor…

We present a simple approach to questions of topological orbit equivalence for actions of countable groups on topological and smooth manifolds. For example, for any action of a countable group $Γ$ on a topological manifold where the fixed sets for any element are contained in codimension two submanifolds, every orbit e…

The Clifford group for 2 qubits is divided into 20 orbits, each with 4608 matrices.

Hamiltonian symplectic actions of tori on compact symplectic manifolds have been extensively studied in the past thirty years, and a number of classifications have been achieved, for instance in the case that the acting torus is $n$-dimensional and the symplectic manifold is $2n$-dimensional. In this case the $n$-dimen…

Let $Σ$ be a connected closed three-manifold, and let $t_Σ$ be the order of the torsion subgroup of $H_1(Σ;\mathbb Z)$. For a contact form $α$ on $Σ$, we denote by $\mathrm{Volume}(α)$ the contact volume of $α$, and by $T_{\min}(α)$ and $T_{\max}(α)$ the minimal period and the maximal period of prime periodic orbits of…

Symplectic classification for a specific type of singularity in integrable systems.

We prove that a maximal totally complex submanifold $N^{2n}$ of the quaternionic projective space $\mathbb{H}\mathbb{P}^n$ ($n\geq 2$) is a parallel submanifold, provided one of the following conditions is satisfied: (1) $N$ is the orbit of a compact Lie group of isometries, (2) the restricted normal holonomy is a prop…

Study closed G2-structures with T3-symmetry, classifying them into types and deriving hypersymplectic structures.

New symmetries discovered in Kepler's orbit family.

We prove a structure theorem for compact aspherical Lorentz manifolds with abundant local symmetry. If M is a compact, aspherical, real-analytic, complete Lorentz manifold such that the isometry group of the universal cover has semisimple identity component, then the local isometry orbits in M are roughly fibers of a f…