arXiv research

We classify the systems of $T$-roots of the flag manifolds $M$ of the exceptional compact simple Lie groups with the second Betti number $b_2(M)\ge2$.

Two exceptional flag manifolds' complex structures are studied, proving rigidity for one.

In the first part of this paper we study geometric formality for generalized flag manifolds, including full flag manifolds of exceptional Lie groups. In the second part we deal with the problem of the classification of invariant almost complex structures on generalized flag manifolds using topological methods.

We construct the Einstein equation for an invariant Riemannian metric on the exceptional full flag manifold $M=G_2/T$. By computing a Gröbner basis for a system of polynomials of multi-variables we prove that this manifold admits exactly two non-Kähler invariant Einstein metrics. Thus $G_2/T$ turns out to be the first …

Study equigeodesics on $G_2$-type flag manifolds, splitting tangent spaces.

We study spin structures on compact simply-connected homogeneous pseudo-Riemannian manifolds (M = G/H, g) of a compact semisimple Lie group G. We classify flag manifolds F = G/H of a compact simple Lie group which are spin. This yields also the classification of all flag manifolds carrying an invariant metaplectic stru…

Study finds 132 complex invariant Einstein metrics on a specific flag manifold and constructs Ricci-flat metrics.

We classify homogeneous reversible Finsler metrics with positive Flag curvature. We show that if G/H admits a G invariant reversible Finsler metric with positive Flag curvature, then up to a few low dimensional spaces, it also admits a G invariant Riemannian metric with positive sectional curvature. For the exceptions,…

We develop matrix models for Grassmann, flag, and Stiefel manifolds.

We consider the action of Anosov subgroups of a semi-simple Lie group on the associated flag manifolds. A systematic approach to construct cocompact domains of discontinuity for this action was given by Kapovich, Leeb and Porti in arXiv:1306.3837. For $Δ$-Anosov representations, we prove that every cocompact domain of …

The main result of this paper is the computation of the Lie superalgebras of holomorphic vector fields on the complex $Π$-symmetric flag supermanifolds, introduced by Yu.I.~Manin. We prove that with one exception any vector field is fundamental with respect to the natural action of the Lie superalgebra $\mathfrak q_n(\…

We present a systematic calculation of the volumes of compact manifolds which appear in physics: spheres, projective spaces, group manifolds and generalized flag manifolds. In each case we state what we believe is the most natural scale or normalization of the manifold, that is, the generalization of the unit radius co…

Let $G$ be a simple compact connected Lie group. We study homogeneous Einstein metrics for a class of compact homogeneous spaces, namely generalized flag manifolds $G/H$ with second Betti number $b_{2}(G/H)=1$. There are 8 infinite families $G/H$ corresponding to a classical simple Lie group $G$ and 25 exceptional flag…

The main result of this paper is the computation of the Lie superalgebras of holomorphic vector fields on complex flag supermanifolds, introduced by Yu.I.Manin. We prove that with several exceptions any holomorphic vector field is fundamental with respect to the natural action of the Lie superalgebra $\mathfrak {gl}_{m…

Let $\mathbb{F}_{Θ}=G/P_{Θ}$ be a flag manifold associated to a non-compact real simple Lie group $G$ and the parabolic subgroup $$. This is a closed subgroup of $G$ determined by a subset $$ of simple restricted roots of $\mathfrak{g}=Lie(G)$. This paper computes the second de Rham cohomology group of $\math…

The moduli space NK of infinitesimal deformations of a nearly Kähler structure on a compact 6-dimensional manifold is described by a certain eigenspace of the Laplace operator acting on co-closed primitive (1,1) forms. Using the Hermitian Laplace operator and some representation theory, we compute the space NK on all 6…

Condition for intersection of real flag manifolds in complex flag manifold.

We show that a harmonic map from a Riemann surface into the exceptional symmetric space $G_2/{\mathrm SO}(4)$ has a $J_2$-holomorphic twistor lift into one of the three flag manifolds of $G_2$ if and only if it is `nilconformal', i.e., has nilpotent derivative. Then we find relationships with almost complex maps from a…

Given an exceptional compact simple Lie group $G$ we describe new left-invariant Einstein metrics which are not naturally reductive. In particular, we consider fibrations of $G$ over flag manifolds with a certain kind of isotropy representation and we construct the Einstein equation with respect to the induced left-inv…

New algorithm computes flag mean and median on flag manifolds.

Minimal dimensions found for flag manifolds embeddings.

Study finds conditions for Kähler-Einstein metrics on flag manifolds.

Study of weighted nonlinear flags in symplectic geometry.

The paper studies Finsler manifolds with a new curvature concept.

The paper examines the geometry of specific submanifolds in flag manifolds.

The paper classifies complex Dirac structures on flag manifolds.

Study nonlinear flags as coadjoint orbits of Hamiltonian diffeomorphisms.

Classifies minimal immersions from $S^2$ into specific flag manifolds.

Study spin chains and sigma models on flag manifolds, calculating spectra and geodesics.

The paper calculates expected distances on partially oriented flag manifolds.

The paper explores curvature positivity on Kähler and quasi-Kähler flag manifolds.

Associated to any finite flag complex L there is a right-angled Coxeter group W_L and a cubical complex Σ_L on which W_L acts properly and cocompactly. Its two most salient features are that (1) the link of each vertex of Σ_L is L and (2) Σ_L is contractible. It follows that if L is a triangulation of S^{n-1}, then Σ_L…

Algorithm to compute cohomology groups of real flag manifolds, proving torsion and Schubert varieties.

Holomorphic structures on quantum flag manifolds uniquely defined.

The study classifies transverse spheres in flag manifolds and finds new examples.

Geometric structures on 5-manifolds from surface group representations of G2'.

Study bifurcations and local rigidity on flag manifolds for Yamabe solutions.

Study and classify totally geodesic submanifolds in nearly Kaehler flag manifold.

Embeds flag manifolds into classical ones, proving rigidity in Kähler geometry.

Geodesic orbit metrics on real flag manifolds identified.

Characterizes invariant spinors on flag manifolds.

The study characterizes real flag manifolds with invariant generalized almost complex structures.

Relates quantum cohomology to tt*-Toda equations for minuscule flag manifolds.

In the present paper we provide a description of complete Calabi-Yau metrics on the canonical bundle of generalized complex flag manifolds. By means of Lie theory we give an explicit description of complete Ricci-flat Kähler metrics obtained through the Calabi ansatz technique. We use this approach to provide several e…

Solves geodesics and Laplace-Beltrami spectrum on flag manifolds.

Researchers prove formulas for flag area measures, extending previous work.

The paper classifies flag manifolds with specific isotropy components and finds conditions for Kähler-like scalar curvature.

Unified PCA framework on flag manifolds for robust data analysis.