arXiv research

Proved dynamical Alekseevskii conjecture in 5D.

If the holonomy representation of an $(n+2)$--dimensional simply-connected Lorentzian manifold $(M,h)$ admits a degenerate invariant subspace its holonomy group is contained in the parabolic group $(\mathbb{R} \times SO(n))\ltimes \mathbb{R}^n$. The main ingredient of such a holonomy group is the SO(n)--projection $G:=…

We define the notion of affine Anosov representations of word hyperbolic groups into the affine group $\mathsf{SO}^0(n+1,n)\ltimes\mathbb{R}^{2n+1}$. We then show that a representation $ρ$ of a word hyperbolic group is affine Anosov if and only if its linear part $\mathtt{L}_ρ$ is Anosov in $\mathsf{SO}^0(n+1,n)$ with …

The holonomy group of an (n+2)-dimensional simply-connected, indecomposable but non-irreducible Lorentzian manifold (M,h) is contained in the parabolic group $(\mathbb{R} \times SO(n))\ltimes \mathbb{R}^n$. The main ingredient of such a holonomy group is the SO(n)--projection $G:=pr_{SO(n)}(Hol_p(M,h))$ and one may ask…

In M-theory vacua with vanishing 4-form F, one can invoke the ordinary Riemannian holonomy H \subset SO(1,10) to account for unbroken supersymmetries n=1, 2, 3, 4, 6, 8, 16, 32. However, the generalized holonomy conjecture, valid for non-zero F, can account for more exotic fractions of supersymmetry, in particular 16<n…

In this article, we provide a necessary and sufficient criterion for proper actions on $\mathbb{H}^{n,n-1}$ in terms of certain special Anosov representations in $\mathsf{SO}(n,n)$. Moreover, we show that affine Anosov representations of any word hyperbolic group in $\mathsf{SO}(n,n-1)\ltimes\mathbb{R}^{2n-1}$ are infi…

The simplest orientifolds of the WZW models are obtained by gauging a Z_2 symmetry group generated by a combined involution of the target Lie group G and of the worldsheet. The action of the involution on the target is by a twisted inversion g \mapsto (ζg)^{-1}, where ζis an element of the center of G. It reverses the …

The blow-down map is studied in Lie algebroid cohomology.

The paper studies bundles over surfaces and constructs invariants.

Extended Siegel-Jacobi upper half-plane geometry studied with invariant metrics.

New method constructs explicit $p$-harmonic functions on Lie groups.

Abstract classifies Lie algebras with complex or symplectic structures.

Let $T$ be a circle and $LT$ be its loop group. Let $\mathcal{M}$ be an infinite dimensional manifold equipped with a nice $LT$-action. We construct an analytic $LT$-equivariant index for $\mathcal{M}$, and justify it in terms of noncommutative geometry. More precisely, we construct a Hilbert space $\mathcal{H}$ consis…

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 classifies reversible and strongly reversible elements in Hermitian isometry groups.

A complex symplectic structure on a Lie algebra $\lie h$ is an integrable complex structure $J$ with a closed non-degenerate $(2,0)$-form. It is determined by $J$ and the real part $Ω$ of the $(2,0)$-form. Suppose that $\lie h$ is a semi-direct product $\lie g\ltimes V$, and both $\lie g$ and $V$ are Lagrangian with re…

Affine deformations of convex cones yield special spacetime structures.

We show that the Frölicher spectral sequence of a complex parallelizable solvmanifold is degenerate at $E_{2}$-term. For a semi-direct product $G=\C^{n}\ltimes_φN$ of Lie-groups with lattice $Γ=Γ^{\prime}\ltimes Γ^{\prime\prime}$ such that $N$ is a nilpotent Lie-group with a left-invariant complex structure and $φ$ is …

We present a fundamental theory of curves in the affine plane and the affine space, equipped with the general-affine groups ${\rm GA}(2)={\rm GL}(2,{\bf R})\ltimes {\bf R}^2$ and ${\rm GA}(3)={\rm GL}(3,{\bf R})\ltimes {\bf R}^3$, respectively. We define general-affine length parameter and curvatures and show how such …

This paper classifies reversible and strongly reversible elements in affine groups.

The paper studies mapping class groups of nontrivial $S^2$ fiber bundles.

Uniform bounds on $SO(2) imes SO(3)$-invariant Ricci solitons on $\mathbb{S}^4$.

We prove complete integrability of the Manakov-type SO(n)-invariant geodesic flows on homogeneous spaces $SO(n)/SO(k_1)\times...\times SO(k_r)$, for any choice of $k_1,...,k_r$, $k_1+...+k_r\le n$. In particular, a new proof of the integrability of a Manakov symmetric rigid body motion around a fixed point is presented…

Prove existence of $SO(3) imes SO(8)$-invariant Einstein metric on $S^3 imes S^7$

Invariant Einstein metrics on generalized Wallach spaces have been classified except $SO(k+l+m)/SO(k)\times SO(l)\times SO(m)$. In this paper, we give a survey on the study of invariant Einstein metrics on generalized Wallach spaces, and prove that there are infinitely many spaces of the type $SO(k+l+m)/SO(k)\times SO(…

Consider the nonstandard embedding of SO(3) into SO(5) given by the 5-dimensional irreducible representation of SO(3), henceforth called SO(3)_\ir. In this note, we study the topology and the differential geometry of 5-dimensional Riemannian manifolds carrying such an SO(3)_\ir structure, i.\,e. with a reduction of the…

The Lie group SO_0(n, 1) has the left-invariant metric coming from the Killing-Cartan form. The maximal compact subgroup SO(n) of the isometry group acts from the left. The geometry of the quotient space of the homogeneous submersion SO_0(n, 1) -> SO(n)\SO_0(n, 1) is investigated. The space is expressed as a warped pro…

We study the stability of symmetric trajectories of a particle on the Lie group $SO(3)$ whose motion is governed by an $SO(3)\times SO(2)$ invariant metric and an $SO(2)\times SO(2)$ invariant potential. Our method is to reduce the number of degrees of freedom at {\em singular} values of the $SO(2)\times SO(2)$ momentu…

The paper studies metrics on Lie groups SO(2) and SO(3) and their integrability.

We develop the theory of equivariant harmonic self-maps of compact cohomogeneity one manifolds and construct new harmonic self-maps of the compact Lie groups SO(4L+2), L >= 1, with degree -3, of SO(8), SO(14) and SO(26) with degree -5 each, of SO(10) with degree -7, and of SO(14) with degree -11 by exhibiting linear so…

For two positive integers m and n, we let ${\mathcal P}_n$ be the open convex cone in ${\mathbb R}^{n(n+1)/2}$ consisting of positive definite n x n real symmetric matrices and let ${\mathbb R}^{(m,n)}$ be the set of all m x n real matrices. In this article, we investigate differential operators on the non-reductive ma…

A contact hypersurface in a Kaehler manifold is a real hypersurface for which the induced almost contact metric structure determines a contact structure. We carry out a systematic study of contact hypersurfaces in Kaehler manifolds. We then apply these general results to obtain classifications of contact hypersurfaces …

The paper explores properties of Lie algebra g2 and related geometric structures.

A nonstandard (maximal) inclusion SO(3) in SO(5) associated with the irreducible representation ρ_5 of SO(3) in R^5 is considered. The topological obstructions for admitting the SO(3) structure on the frame bundle over 5-manifold are investigated. The necessary and sufficient conditions are formulated.

The paper classifies topological holonomy groups in $SO(3)$.

Established a version of the Atiyah-Floer conjecture for SO(3)-bundles.

The paper studies SDP feasibility and sos ranks for specific polynomials.

The Riemannian submersion $π: \text{SO}_0(1,n) \to \mathbb{H}^n$ is a principal bundle and its fiber at $π(e)$ is the imbedding of $\text{SO}(n)$ into $\text{SO}_0(1,n)$, where $e$ is the identity of both $\text{SO}_0(1,n)$ and $\text{SO}(n)$. In this study, we associate a curve, starting from the identity, in $\…

We consider the nonstandard inclusion of SO(3) in SO(5) associated with a 5-dimensional irreducible representation. The tensor $Υ$ representing this reduction is found to be given by a ternary symmetric form with special properties. A 5-dimensional manifold $(M,g,Υ)$ with Riemannian metric $g$ and ternary form generate…

We classify all connected subgroups of SO(2,n) that act irreducibly on $\R^{2,n}$. Apart from $SO_0(2,n)$ itself these are $U(1,n/2)$, $SU(1,n/2)$, if $n$ even, $S^1\cdot SO(1,n/2)$ if $n$ even and $n\ge 2$, and $SO_0(1,2)$ for $n=3$. Our proof is based on the Karpelevich Theorem and uses the classification of totally …

The article classifies 6D flat solvmanifolds by analyzing conjugacy classes of matrices.

A left-invariant sub-Riemannian metric $d$ on the shortened Lorentz group $SO_0(2,1)$ under the condition that $d$ is right-invariant relative to the orthogonal Lie subgroup $1\otimes SO(2)$ is studied. The distance between arbitrary two elements, the cut locus (as the union of the subgroup $1\otimes SO(2)$ with the an…

Symmetric spaces have unique spectra under certain group actions.

Researchers create explicit p-harmonic functions on specific symmetric spaces.

The paper calculates R and Racah matrices for SO(5) and finds Kauffman polynomials.

We study the geometry of a $G$-structure $P$ inside the oriented orthonormal frame bundle ${\rm SO}(M)$ over an oriented Riemannian manifold $M$. We assume that $G$ is connected and closed, so the quotient ${\rm SO}(n)/G$, where $n=\dim M$, is a normal homogeneous space and we equip ${\rm SO}(M)$ with the natural Riema…

The authors find geodesics, shortest arcs, diameter, cut locus, and conjugate sets for left-invariant sub-Riemannian metric on the Lie group SO(3), under condition that the metric is right-invariant relative to the Lie subgroup $SO(2)\subset SO(3)$.

Concrete proof that SO(n,1) is not a T-group.