arXiv research

Paper presents a novel orbit determination method for spacecraft clusters.

The paper analyzes orbits of integer tuples using braid diagrams.

Classifies pseudo-Anosov flows on 3-manifolds up to orbit equivalence.

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…

Let G be a real compact connected simple Lie group, and g its Lie algebra. We study the problem of determining, from root data, when a sum of adjoint orbits in g, or a product of conjugacy classes in G, contains an open set. Our general methods allow us to determine exactly which sums of adjoint orbits in su(m) and pro…

For a representation of a finite group $G$ on a complex vector space $V$ we determine when a holomorphic $\binom{p}{q}$-tensor field on the principle stratum of the orbit space $V/G$ can be lifted to a holomorphic $G$-invariant tensor field on $V$. This extends also to connections. As a consequence we determine those h…

Study on orbit spaces with curvature bounds.

Classifies actions on Minkowski space up to orbit equivalence.

Lie groupoids and their orbit spaces are linked through equivalence classes.

Study orbits in right triangles, deducing periodic billiard paths and classifying orbit closures.

New insights into pseudo-Anosov flows with special periodic orbits.

We introduce the notion of a weakly reflective submanifold, which is an austere submanifold with a certain global condition, and study its fundamental properties. Using these, we determine weakly reflective orbits and austere orbits of s-representations.

We give a necessary and sufficient condition for orbits of commutative Hermann actions and actions of the direct product of two symmetric subgroups on compact Lie groups to be biharmonic in terms of symmetric triad with multiplicities. By this criterion, we determine all the proper biharmonic submanifolds in irreducibl…

Let $\mathcal{J}^1$ be the real form of complex simple Jordan algebra with the automorphism group $G$ of type $F_{4(-20)}$. Explicitly, we give the orbit decomposition of $\mathcal{J}^1$ under the action of $G$ and determine the Lie group structure of stabilizer for each $G$-orbit on $\mathcal{J}^1$.

Geodesic graphs for special Finsler metrics on spheres are studied.

A classical result says that a free action of the circle $\Bbb{S}^1$ on a topological space $X$ is geometrically classified by the orbit space $B$ and by a cohomological class ${H}^{^{2}}{(B,\Bbb{Z})}$, the Euler class. When the action is not free we have a difficult open question: $Π$ : "Is the space $X$ determined by…

The paper characterizes contact 3-manifolds with closed Reeb orbits.

Reconstruct flows from their orbit spaces using group actions.

Closed and broken electromagnetic orbits in Kerr-Newman spacetime

In this paper we give a classification of closed and connected Lie groups, up to conjugacy in $Iso({\bf adS_3})$, acting by cohomogeneity one on the three dimensional anti de sitter space ${\bf adS_3}$. Then we determine causal characters of the orbits and the orbit spaces, up to homeomorphism, in both cases, proper an…

We study the structure of finite quandles in terms of subquandles. Every finite quandle $Q$ decomposes in a natural way as a union of disjoint $Q$-complemented subquandles; this decomposition coincides with the usual orbit decomposition of $Q$. Conversely, the structure of a finite quandle with a given orbit decomposit…

The paper studies mean curvature flows on specific orbits of Hermann actions.

A Lie group $G$ naturally acts on its Lie algebra $\gg$, called the adjoint action. In this paper, we determine the orbit types of the compact exceptional Lie group $G_2$ in its Lie algebra $\gg_2$. As results, the group $G_2$ has four orbit types in the Lie algebra $\gg_2$ as $$ G_2/G_2, \quad G_2/(U(1) \times U(1)), …

A geodesic orbit manifold (GO manifold) is a Riemannian manifold (M,g) with the property that any geodesic in M is an orbit of a one-parameter subgroup of a group G of isometries of (M,g). The metric g is then called a G-GO metric in M. For an arbitrary compact homogeneous manifold M=G/H, we simplify the general proble…

Classifies homogeneous hypersurfaces in specific 4D geometries.

Computer program finds flower-like periodic orbits in Kepler-Heisenberg problem.

Classifies conjugation orbits of hyperbolic elements in various groups and surfaces.

Study $Sp(n)$-orbits in complex and $Σ$-complex subspaces of Hermitian quaternionic vector spaces.

The paper solves orbital integrals on Lorentzian symmetric spaces.

Study open orbits in causal flag manifolds with applications in AQFT.

We study polar orbitopes, i.e. convex hulls of orbits of a polar representation of a compact Lie group. The face structure is studied by means of the gradient momentum map and it is shown that every face is exposed and is again a polar orbitope. Up to conjugation the faces are completely determined by the momentum poly…

Study of laminations for pseudo-Anosov flows on three-manifolds.

Let G be a (real or complex) linear reductive algebraic group acting on an affine variety V. Let W be a subvariety. In this work we study how the G-orbits intersect W. We develop a criterion to determine when the intersection can be described as a finite union of orbits of a reductive subgroup. The conditions of the cr…

In this paper we classify, up to orbit equivalence, cohomogeneity one actions of connected closed Lie subgroups of $U(1,n)$ on the $(2n+1)$-dimensional anti de Sitter spacetime $AdS^{2n+1}$. We also give some new examples of nonproper cohomogeneity one actions on $AdS^{n+1}$ and determine parabolic Lie subgroups of $SO…

Zero-energy orbits in the Kepler-Heisenberg problem are self-similar and stratify into three families.

The paper studies cohomogeneity one actions on pseudo-Euclidean space and identifies unique orbit structures.

Proposes a model-free control method for chaotic systems using deep Q-learning.

We study mapping class group orbits of homotopy and isotopy classes of curves with self-intersections. We exhibit the asymptotics of the number of such orbits of curves with a bounded number of self-intersections, as the complexity of the surface tends to infinity. We also consider the minimal genus of a subsurface tha…

We prove that for a weakly exact magnetic system on a closed connected Riemannian manifold, almost all energy levels contain a closed orbit. More precisely, we prove the following stronger statements. Let $(M,g)$ denote a closed connected Riemannian manifold and $σ$ a weakly exact 2-form. Let $φ_{t}$ denote the magneti…

For a principal $\rmSU(n)$-bundle over a compact manifold of dimension $2,3,4$, we determine the orbit types of the action of the gauge group on the space of connections modulo pointed local gauge transformations. We find that they are given by Howe subgroups of $\rmSU(n)$ for which a certain characteristic equation is…

Let J be the exceptional Jordan algebra over R and J^C its complexification. Then the simply connected compact exceptional Lie group F_4 acts on J and F_4 has three orbit types which are F_4/F_4, F_4/Spin(9), F_4/Spin(8). Similarly the simply connected compact exceptional Lie group E_6 acts on J^C and E_6 has five orbi…

We study the quandle counting invariant for a certain family of finite quandles with trivial orbit subquandles. We show how these invariants determine the linking number of classical two-component links up to sign.

We emphasize some properties of coherent state groups, i.e. groups whose quotient with the stationary groups, are manifolds which admit a holomorphic embedding in a projective Hilbert space. We determine the differential action of the generators of the representation of coherent state groups on the symmetric Fock space…

The paper proves rigidity results for Anosov flows and their orbit equivalences.

Flat surfaces that correspond to meromorphic $1$-forms or to meromorphic quadratic differentials containing poles of order two and higher are surfaces of infinite area. We classify groups that appear as Veech groups of translation surfaces with poles. We characterize those surfaces such that their $GL^{+}(2,\mathbb{R})…

We prove similar theorems concerning the structure of bundles involving complements of fiber-type hyperplane arrangements and orbit configuration spaces. These results facilitate analysis of the fundamental groups of these spaces, which may be viewed as generalizations of the Artin pure braid group. In particular, we r…

Researchers describe Casimir functions for 3- and 4-step nilpotent Lie groups.

In this paper, we study biharmonic hypersurfaces in Einstein manifolds. Then, we determine all the biharmonic hypersurfaces in irreducible symmetric spaces of compact type which are regular orbits of commutative Hermann actions of cohomogeneity one.