arXiv research

In this paper, we study normal homogeneous Finsler spaces. We first define the notion of a normal homogeneous Finsler space, using the method of isometric submersion of Finsler metrics. Then we study the geometric properties. In particular, we establish a technique to reduce the classification of normal homogeneous Fin…

Survey on positively curved homogeneous Finsler spaces.

The paper extends a classical result to positively curved homogeneous spaces.

Ricci flow preserves positive sectional curvature on homogeneous spheres

Paper classifies positively curved δ-homogeneous Finsler metrics.

Establishes metrics with positive 2nd intermediate Ricci curvature on products of curved spaces.

In this paper, we use the flag curvature formula for homogeneous Finsler spaces in our previous work to classify odd dimensional smooth coset spaces admitting positively curved reversible homogeneous Finsler metrics. We will show that the most features of L. Bérard-Bergery's classification results for odd dimensional p…

The paper proves rigidity results for non-positively curved homogeneous Finsler metrics.

In this paper, we introduce the flag-wise positively curved condition for Finsler spaces (the (FP) Condition), which means that in each tangent plane, we can find a flag pole in this plane such that the corresponding flag has positive flag curvature. Applying the Killing navigation technique, we find a list of compact …

We examine homogeneous metrics on spheres and determine which ones have positive sectional curvature. The answer is subtle and surprisingly difficult to prove. In some cases we also determine their pinching constants. This completes the classification of all homogeneous metrics with positive curvature (apart from one s…

A Finsler space $(M,F)$ is called flag-wise positively curved, if for any $x\in M$ and any tangent plane $\mathbf{P}\subset T_xM$, we can find a nonzero vector $y\in \mathbf{P}$, such that the flag curvature $K^F(x,y, \mathbf{P})>0$. Though compact positively curved spaces are very rare in both Riemannian and Finsler g…

We study the topology of the 13 dimensional positively curved Bazaikin spaces. We show that there is only one such manifold which is homotopy equivalent to a homogeneous space, the so called Berger space. This is in contrast to the case of the 7 dimensional positively curved Eschenburg spaces. In addition, we compute t…

New proof for curved 3-cohom manifold rational ellipticity.

Linear inequality found for certain curved spaces.

The paper classifies compact homogeneous Finsler manifolds with positive flag curvature.

New manifolds found with almost everywhere positive curvature.

The study finds that certain curved manifolds can be mapped to symmetric spaces.

Study verifies Homogeneity Conjecture for three odd-dimensional spheres in positive curvature.

As was recently observed by M. Xu and J. Wolf, there is a gap in Berard Bergery's classification of odd dimensional positively curved homogeneous spaces. Since this classification has been used in other papers as well, we give a modern, complete and self contained proof (in odd as well as even dimensions), confirming t…

The known manifolds of positive sectional curvature are either homogeneous spaces or biquotients, i.e. quotients of a compact Lie group by a group acting on the left and right simultaneously. The full isometry group of the homogeneous metrics of positive curvature were determined by K.Shankar. Here we determine the iso…

Let $G$ be a compact Lie group acting isometrically on a compact Riemannian manifold $M$ with nonempty fixed point set $M^G$. We say that $M$ is fixed-point homogeneous if $G$ acts transitively on a normal sphere to some component of $M^G$. Fixed-point homogeneous manifolds with positive sectional curvature have been c…

A compact Riemannian homogeneous space $G/H$, with a bi--invariant orthogonal decomposition $\mathfrak{g}=\mathfrak{h}+\mathfrak{m}$ is called positively curved for commuting pairs, if the sectional curvature vanishes for any tangent plane in $T_{eH}(G/H)$ spanned by a linearly independent commuting pair in $\mathfrak{…

New results affirmatively answer the stable converse soul question for many curved spaces.

Formula derived for Laplace-Beltrami spectrum on homogeneous spaces.

No stable discrete maps into certain curved spaces exist.

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

Generalizes SRVF to curves in homogeneous spaces for efficient distance computation.

This paper is devoted to the study of the evolution of positively curved metrics on the Wallach spaces $SU(3)/T_{\max}$, $Sp(3)/Sp(1)\times Sp(1)\times Sp(1)$, and $F_4/Spin(8)$. We prove that for all Wallach spaces, the normalized Ricci flow evolves all generic invariant Riemannian metrics with positive sectional curv…

Let $M$ be a hyperkähler manifold with $b_2(M)\geq 5$. We improve our earlier results on the Morrison-Kawamata cone conjecture by showing that the Beauville-Bogomolov square of the primitive MBM classes (i.e. the classes whose orthogonal hyperplanes bound the Kähler cone in the positive cone, or, in other words, the cl…

This is a slightly altered version of the authors thesis from 2014. In the first main part we show that the quotient space of a compact, simply connected and nonnegatively curved Riemannian 4-manifold by an effective, isometric circle-action admits an approximation in Gromov-Hausdorff topology by smooth, positively cur…

We show how to specify preferred parameterisations on a homogeneous curve in an arbitrary homogeneous space. We apply these results to limit the natural parameters on distinguished curves in parabolic geometries.

The paper extends square root velocity framework to curves in homogeneous spaces.

Enhances the Bishop-Gromov theorem for curved spaces, especially at late times.

Abstract: Characterizes spaces with positive scalar curvature.

Examples of almost-positively and quasi-positively curved spaces of the form M=H((G,h)xF) were discovered recently. Here, h is a left-invariant metric on a compact Lie group G, F is a compact Riemannian manifold on which the subgroup H of G acts isometrically on the left, and M is the orbit space of the diagonal left a…

Study distinguishes components of moduli spaces for certain metrics.

Characterizes equigeodesics on specific homogeneous spaces.

The current paper is devoted to the study of integral curves of constant type in parabolic homogeneous spaces. We construct a canonical moving frame bundle for such curves and give the criterium when it turns out to be a Cartan connection. Generalizations to parametrized curves, to higher-dimensional submanifolds and t…

New criterion for cylinder stability in curved spaces.

The study preserves Ricci curvature positivity on homogeneous spaces under the normalized Ricci flow.

Classifies manifolds with quasipositive curvature.

Ricci flow on certain homogeneous spaces creates metrics with positive curvature.

Study on triharmonic curves in 3D spaces, proving their existence and classification.

The $L^p$-cohomology in degree 1 of Riemannian homogeneous spaces is computed. It turns out that reduced cohomology does not vanish exactly for spaces quasiisometric to negatively curved homogeneous spaces.

Positive simplicial volume implies locally symmetric space structure.

The paper extends Heintze-Kobayashi-Wolf theory to negatively curved homogeneous Finsler manifolds.

We prove local existence of complex-valued harmonic morphisms from any Riemannian homogeneous spaces of positive curvature, except the Berger space Sp(2)/SU(2).

Rollings of reductive homogeneous spaces are studied using intrinsic curves.