arXiv research

Smooth manifolds from locally homogeneous spaces.

New proof shows compact homogeneous LCK manifolds are Vaisman.

New proof classifies homogeneous 3-Sasakian and quaternionic Kähler manifolds.

Researchers found all homogeneous structure tensors on two specific 3D manifolds.

In previous papers, a fundamental affine method for studying homogeneous geodesics was developed. Using this method and elementary differential topology it was proved that any homogeneous affine manifold and in particular any homogeneous pseudo-Riemannian manifold admits a homogeneous geodesic through arbitrary point. …

Study finds all 4D neutral manifolds.

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

The paper proves rigidity properties of holomorphic isometries into homogeneous Kähler manifolds.

The paper studies Randers and $(α,β)$ equigeodesics on compact homogeneous manifolds.

We study in this paper previously defined by V.N. Berestovskii and C.P. Plaut $δ$-homogeneous spaces in the case of Riemannian manifolds. Every such manifold has non-negative sectional curvature. The universal covering of any $δ$-homogeneous Riemannian manifolds is itself $δ$-homogeneous. In turn, every simply connecte…

Classifies totally geodesic submanifolds in specific geometric spaces.

We prove that a four-dimensional Lorentzian manifold that is curvature homogeneous of order 3, or $\CH_3$ for short, is necessarily locally homogeneous. We also exhibit and classify four-dimensional Lorentzian, $\CH_2$ manifolds that are not homogeneous.

In this paper, we provide a systematic and constructive description of Vaisman structures on certain principal elliptic bundles over complex flag manifolds. From this description we explicitly classify homogeneous l.c.K. structures on compact homogeneous Hermitian manifolds using elements of representation theory of co…

Study reveals structure of isometry group for specific manifolds.

Minimal number of geodesics in Finsler manifolds with indefinite Killing form is at least four.

For any k which is at least 2, we exhibit complete k-curvature homogeneous neutral signature pseudo-Riemannian manifolds which are not k+1-affine curvature homogeneous, and hence not locally homogeneous. All the local scalar Weyl invariants of these manifolds vanish. These manifolds are Ricci flat, Osserman, and Ivanov…

We will discuss in this paper homogeneous locally conformally Keahler (or shortly homogeneous l.c.K.) manifolds and locally homogeneous l.c.K. manifolds from various aspects of study in the field of l.c.K. geometry. We will provide a survey of known results along with some new results and observations; in particular we…

Local Sasaki-Ricci solitons are $η$-Einstein in certain fiber products of homogeneous Sasakian manifolds.

k-Curvature homogeneous three-dimensional Walker metrics are described for k=0,1,2. This allows a complete description of locally homogeneous three-dimensional Walker metrics, showing that there exist exactly three isometry classes of such manifolds. As an application one obtains a complete description of all locally h…

Finite time for Ricci flow on certain manifolds.

In spin geometry, traceless cyclic homogeneous Riemannian manifolds equipped with a homogeneous spin structure can be viewed as the simplest manifolds after Riemannian symmetric spin spaces. In this paper, we give some characterizations and properties of cyclic and traceless cyclic homogeneous Riemannian manifolds and …

New class of complex manifolds defined, properties studied.

Develops theory of homogeneous statistical manifolds and classifies Lie groups.

The 2-parameter family of certain homogeneous Lorentzian 3-manifolds which includes Minkowski 3-space and anti-de Sitter 3-space is considered. Each homogeneous Lorentzian 3-manifold in the 2-parameter family has a solvable Lie group structure with left invariant metric. A generalized integral representation formula wh…

The study classifies homogeneous Sasaki manifolds over quaternionic Kähler spaces.

Holomorphic structures on Oeljeklaus-Toma manifolds are shown to be locally homogeneous.

For k at least 2, we exhibit complete k-curvature homogeneous neutral signature pseudo-Riemannian manifolds which are not locally affine homogeneous (and hence not locally homogeneous). The curvature tensor of these manifolds is modeled on that of an indecomposible symmetric space. All the local scalar Weyl curvature i…

We construct a family of balanced signature pseudo-Riemannian manifolds, which arise as hypersurfaces in flat space, that are curvature homogeneous, that are modeled on a symmetric space, and that are not locally homogeneous.

Study homogeneous Lorentzian manifolds under reductive Lie groups, reducing descriptions to semisimple groups.

Study geodesic complexity in homogeneous Riemannian manifolds.

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…

We study a family of 3-dimensional Lorentz manifolds. Some members of the family are 0-curvature homogeneous, 1-affine curvature homogeneous, but not 1-curvature homogeneous. Some are 1-curvature homogeneous but not 2-curvature homogeneous. All are 0-modeled on indecomposible local symmetric spaces. Some of the members…

We classify all compact simply connected homogeneous CR manifolds $M$ of codimension one and with non-degenerate Levi form up to CR equivalence. The classification is based on our previous results and on a description of the maximal connected compact group $G(M)$ of automorphisms of $M$. We characterize also the standa…

Survey updates knowledge on homogeneous Einstein spaces.

The author reviews his results on locally compact homogeneous spaces with inner metric, in particular, homogeneous manifolds with inner metric. The latter are isometric to homogeneous (sub-)Finslerian manifolds; under some additional conditions they are isometric to homogeneous (sub)-Riemannian manifolds. The class $Ω$…

Strong formal properties for toric and homogeneous Kähler manifolds.

In this note we study globally homogeneous Riemannian quotients $Γ\backslash (M,ds^2)$ of homogeneous Riemannian manifolds $(M,ds^2)$. The Homogeneity Conjecture is that $Γ\backslash (M,ds^2)$ is (globally) homogeneous if and only if $(M,ds^2)$ is homogeneous and every $γ\in Γ$ is of constant displacement on $(M,ds^2)$…

Free groups can be end homogeneity groups of 3-manifolds.

The 2-parameter family of certain homogeneous Lorentzian 3-manifolds which includes Minkowski 3-space, de Sitter 3-space, and Minkowski motion group is considered. Each homogeneous Lorentzian 3-manifold in the 2-parameter family has a solvable Lie group structure with left invariant metric. A generalized integral repre…

We show that generalized plane wave manifolds are complete, strongly geodesically convex, Osserman, Szabo, and Ivanov-Petrova. We show their holonomy groups are nilpotent and that all the local Weyl scalar invariants of these manifolds vanish. We construct isometry invariants on certain families of these manifolds whic…

Proves Yau-Tian-Donaldson conjecture for toric manifolds and bundles.

We exhibit 3 families of complete curvature homogeneous pseudo-Riemannian manifolds which are modeled on irreducible symmetric spaces and which are not locally homogeneous. All of the manifolds have nilpotent Jacobi operators; some of the manifolds are, in addition, Jordan Osserman and Jordan Ivanov-Petrova.

We present the construction of a large class of homogeneous KT, HKT and QKT manifolds, $G/K$, using an invariant metric on $G$ and the canonical connection. For this a decomposition of the Lie algebra of $G$ is employed, which is most easily described in terms of colourings of Dynkin diagrams of simple Lie algebras. KT…

We study in this paper three natural notions of convergence of homogeneous manifolds, namely infinitesimal, local and pointed, and their relationship with a fourth one, which only takes into account the underlying algebraic structure of the homogeneous manifold and is indeed much more tractable. Along the way, we intro…

New product manifolds can have non-negative curvature.

Extends rigidity results to non-homogeneous manifolds.

Smooth Schubert varieties in rational homogeneous manifolds of Picard number 1 are horospherical varieties. We characterize standard embeddings of smooth Schubert varieties in rational homogeneous manifolds of Picard number 1 by means of varieties of minimal rational tangents. In particular, we mainly consider nonhomog…

The study characterizes compact homogeneous manifolds with Bismut parallel torsion.