arXiv research

An infinite family of distinct harmonic morphisms with minimal circle fibers from the 7-dimensional homogeneous Allof-Wallach spaces of positive curvature onto the 6-dimensional flag manifolds is given.

Study Palais-Smale sequences for Ricci curvature on homogeneous spaces.

Study Ricci flow on specific homogeneous spaces.

This paper describes metrics with varying curvature properties in a specific manifold.

We obtain a complete description of the moduli spaces of homogeneous metrics with strongly positive curvature on the Wallach flag manifolds $W^6$, $W^{12}$ and $W^{24}$, which are respectively the manifolds of complete flags in $\mathbb C^3$, $\mathbb H^3$ and $\mathbb{Ca}^3$. Together with our earlier work, this concl…

In this paper, we use the technique of Finslerian submersion to deduce a flag curvature formula for homogeneous Finsler spaces. Based on this formula, we give a complete classification of even-dimensional smooth coset spaces $G/H$ admitting $G$-invariant Finsler metrics with positive flag curvature. It turns out that t…

Study shows metrics on certain manifolds lose positive curvature under Ricci flow.

The paper explores symplectic connections on homogeneous spaces, finding a unique invariant connection.

Consider a compact Lie group $G$ and a closed Lie subgroup $H<G$. Let $\mathcal M$ be the set of $G$-invariant Riemannian metrics on the homogeneous space $M=G/H$. By studying variational properties of the scalar curvature functional on $\mathcal M$, we obtain an existence theorem for solutions to the prescribed Ricci …

Study harmonicity of normal almost contact structures on Riemannian manifolds.

We give an overview of progress on homogeneous Einstein metrics on large classes of homogeneous manifolds, such as generalized flag manifolds and Stiefel manifolds. The main difference between these two classes of homogeneous spaces is that their isotropy representation does not contain/contain equivalent summands. We …

Formula for Lichnerowicz Laplacian on invariant metrics, deducing stability of Einstein manifolds.

We study geodesics in generalized Wallach spaces which are expressed as orbits of products of three exponential terms. These are homogeneous spaces $M=G/K$ whose isotropy representation decomposes into a direct sum of three submodules $\frak{m}=\frak{m}_1\oplus\frak{m}_2\oplus\frak{m}_3$, satisfying the relations $[\fr…

We consider invariant Einstein metrics on the quaternionic Stiefel manifolds $V_p\mathbb{H} ^n$ of all orthonormal $p$-frames in $\mathbb{H}^n$. This manifold is diffeomorphic to the homogeneous space $\mathrm{Sp}(n) / \mathrm{Sp}(n-p)$ and its isotropy representation contains equivalent summands. We obtain new Einstei…

Indecomposable symmetric Lorentzian manifolds of non-constant curvature are called Cahen-Wallach spaces. Their isometry classes are described by continuous families of real parameters. We derive necessary and sufficient conditions for the existence of compact quotients of Cahen-Wallach spaces in terms of these paramete…

This paper classifies 13D manifolds based on Bazaikin spaces.

Study on conformal transformations of Cahen-Wallach spaces, focusing on fixed points and discontinuous groups.

The study characterizes and verifies equivariant embeddings of symmetric Kählerian manifolds.

Upper bounds on Einstein metrics on homogeneous spaces.

Study second-order obstruction to nearly $G_2$ structure deformations.

We classify generalized Wallach spaces which are g.o. spaces. We also investigate homogeneous geodesics in generalized Wallach spaces for any given invariant Riemannian metric and we give some examples.

We study geodesics of the form $γ(t)=π(\exp(tX)\exp(tY))$, $X,Y\in \fr{g}=\operatorname{Lie}(G)$, in homogeneous spaces $G/K$, where $π:G\rightarrow G/K$ is the natural projection. These curves naturally generalise homogeneous geodesics, that is orbits of one-parameter subgroups of $G$ (i.e. $γ(t)=π(\exp (tX))$, $X\in …

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

We introduce a geometric invariant that we call the index of symmetry, which measures how far is a Riemannian manifold from being a symmetric space. We compute, in a geometric way, the index of symmetry of compact naturally reductive spaces. In this case, the so-called leaf of symmetry turns out to be of the group type…

New solutions found for G2 system on squashed manifolds.

In this paper, we present the classification of generalized Wallach spaces and discuss some related problems.

Study of a 3D system on Wallach spaces, finding interrelations with invariant metrics.

We classify two-symmetric Lorentzian manifolds using methods of the theory of holonomy groups. These manifolds are exhausted by a special type of pp-waves and, like the symmetric Cahen-Wallach spaces, they have commutative holonomy.

Characterizes equigeodesics on specific homogeneous spaces.

Curved metrics on Wallach spaces bounded by curves under flow.

For gauge groups $U(1)$ and $SO(3)$ we classify invariant $G_2$-instantons for homogeneous coclosed $G_2$-structures on Aloff-Wallach spaces $X_{k,l}$. As a consequence, we give examples where $G_2$-instantons can be used to distinguish between different strictly nearly parallel $G_2$-structures on the same Aloff-Walla…

The article constructs Spin(7) metrics with Aloff--Wallach spaces as orbits.

Third-order symmetric Lorentzian manifolds, i.e. Lorentzian manifold with zero third derivative of the curvature tensor, are classified. These manifolds are exhausted by a special type of pp-waves, they generalize Cahen-Wallach spaces and second-order symmetric Lorentzian spaces.

New algorithm computes flag mean and median on flag manifolds.

Study finds bifurcation and local rigidity points for solutions to the Yamabe problem on Aloff-Wallach Spaces.

Minimal dimensions found for flag manifolds embeddings.

Researchers compute the ν-invariant for specific G2-structures on Aloff-Wallach spaces.

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

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.

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(…

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.

Normalized Ricci flow preserves positivity of Ricci curvature on some generalized Wallach spaces.

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

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