arXiv research

Cohomogeneity-one actions on symmetric spaces of mixed type

We study isometric cohomogeneity one actions on the (n+1)-dimensional Minkowski space up to orbit-equivalence. We give examples of isometric cohomogeneity one actions on the Minkowski space whose orbit spaces are non-Hausdorff. We show that there exist isometric cohomogeneity one actions on the Minkowski space which ar…

An isometric action of a Lie group on a Riemannian manifold is of cohomogeneity one if the corresponding orbit space is one-dimensional. In this article we develop a conceptual approach to the classification of cohomogeneity one actions on Riemannian symmetric spaces of noncompact type in terms of orbit equivalence. As…

Complete classification of symmetric spaces actions.

New structural result classifies actions on symmetric spaces.

The aim of this paper is to classify the cohomogeneity one conformal actions on the 3-dimensional Einstein universe $Ein^{1;2}$ , up to orbit equivalence. In a recent paper [21], we studied the unique (up to conjugacy) irreducible action of $PSL(2; \mathbb{R})$ on $Ein^{1;2}$ and we showed that the action is of cohomog…

In this paper, we provide a complete classification for all the isometric cohomogeneity one actions on unit spheres. Using this theory, we can very easily classify all the isometric cohomogeneity one actions on the Riemannian symmetric spaces $\mathbb{C}\mathrm{P}^{m-1}$ and $\mathbb{H}\mathrm{P}^{m-1}$.

The aim of this paper is to study cohomogeneity one isometric linear actions on the $p+q$-dimensional pseudo-Euclidean space $\mathbb{R}^{p,q}$. It is proved that the natural isometric action of the nilpotent factor of an Iwasawa decomposition of $SO(p,q)$ is not of cohomogeneity one. The orbits of cohomogeneity one ac…

Abstract: Characterizes spaces with positive scalar curvature.

We compute the rational Borel equivariant cohomology ring of a cohomogeneity-one action of a compact Lie group.

In this paper we classify, up to orbit equivalence, cohomogeneity one actions of connected closed Lie subgroups of $U(1,n)$ on the $(2n+1)$-dimensional anti de Sitter spacetime $AdS^{2n+1}$. We also give some new examples of nonproper cohomogeneity one actions on $AdS^{n+1}$ and determine parabolic Lie subgroups of $SO…

We show that if $G\times M \to M$ is a cohomogeneity one action of a compact connected Lie group $G$ on a compact connected manifold $M$ then $H^*_G(M)$ is a Cohen-Macaulay module over $H^*(BG)$. Moreover, this module is free if and only if the rank of at least one isotropy group is equal to the rank of $G$. We deduce …

Study GKM actions on special manifolds with interval orbit spaces.

Classification results are given for (i) compact quaternionic Kähler manifolds with a cohomogeneity-one action of a semi-simple group, (ii) certain complete hyperKähler manifolds with a cohomogeneity-two action of a semi-simple group preserving each complex structure, (iii) compact 3-Sasakian manifolds which are cohomo…

We classify closed, simply-connected cohomogeneity-one Alexandrov spaces in dimensions $5$, $6$ and $7$. We show that every closed, simply-connected smooth $n$-orbifold, $2\leq n\leq 7$ with a cohomogeneity one action is equivariantly homeomorphic to a smooth good orbifold of cohomogeneity one.

Classifies polar actions on 3D homogeneous spaces.

We classify, up to orbit equivalence, the cohomogeneity one actions on the noncompact duals of the symmetric spaces G_2, SU_3 and the real oriented two-plane Grassmannians.

The Ambrose-Singer theorem is extended to cohomogeneity one Riemannian manifolds.

Study cohomogeneity one RCD-spaces, proving structural results and constructing new examples.

We construct examples of cohomogeneity one special Lagrangian submanifolds in the cotangent bundle over the complex projective space, whose Calabi-Yau structure was given by Stenzel. For each example, we describe the condition of special Lagrangian as an ordinary differential equation. Our method is based on a moment m…

Study finds metrics with positive intermediate Ricci curvature on specific low-dimensional manifolds.

The aim of this paper is to classify the cohomogeneity one conformal actions on the three-dimensional essential Riemannian spaces, up to orbit equivalence. Among other results, the representations of all connected Lie groups acting with cohomogeneity one or zero within the full conformal group of a given three-dimensio…

New examples of hypersurfaces found in quaternionic hyperbolic spaces.

Proves Penrose inequality for cohomogeneity one initial data sets.

Classifies totally geodesic submanifolds and polar actions on Stiefel manifolds.

Study on Einstein metrics on complex projective spaces with specific group actions.

The aim of this paper is to classify cohomogeneity one isometric actions on the 4-dimensional Minkowski space $\mathbb{R}^{3,1}$, up to orbit equivalence. Representations, up to conjugacy, of the acting groups in $O(3,1)\ltimes \mathbb{R}^{3,1}$ are given in both cases, proper and non-proper actions. When the action is…

We obtain a structure theorem for closed, cohomogeneity one Alexandrov spaces and we classify closed, cohomogeneity one Alexandrov spaces in dimensions 3 and 4. As a corollary, we obtain the classification of closed, $n$-dimensional, cohomogeneity one Alexandrov spaces admitting an isometric $T^{n-1}$ action. In contra…

In these notes, after an introduction to toric Kahler geometry, we present Calabi's family of U(n)-invariant extremal Kahler metrics in symplectic action-angle coordinates and show that it actually contains, as particular cases, many interesting cohomogeneity one examples of constant scalar curvature.

Study on solutions of Yamabe-type equations on projective spaces.

Study on special Hermitian metrics on cohomogeneity one manifolds.

Proves Yau-Tian-Donaldson conjecture for cohomogeneity one manifolds.

Classifies timelike translating solitons in Minkowski space.

Study on $G_{2}$ actions on symmetric spaces, focusing on orbit properties.

Researchers classify curvature homogeneous metrics on 4D manifolds.

In this paper we give a classification of closed and connected Lie groups, up to conjugacy in $Iso({\bf adS_3})$, acting by cohomogeneity one on the three dimensional anti de sitter space ${\bf adS_3}$. Then we determine causal characters of the orbits and the orbit spaces, up to homeomorphism, in both cases, proper an…

We study gradient Ricci solitons with maximal symmetry. First we show that there are no non-trivial homogeneous gradient Ricci solitons. Thus the most symmetry one can expect is an isometric cohomogeneity one group action. Many examples of cohomogeneity one gradient solitons have been constructed. However, we apply the…

We classify, up to orbit equivalence, all cohomogeneity one actions on the hyperbolic planes over the complex, quaternionic and Cayley numbers, and on the complex hyperbolic spaces of dimension greater than two. For the quaternionic hyperbolic spaces of dimension greater than two we reduce the classification problem to…

Investigates harmonic self-maps' stability on cohomogeneity one manifolds.

We classify the polar actions on the complex hyperbolic plane up to orbit equivalence. Apart from the trivial and transitive polar actions, there are five polar actions of cohomogeneity one and four polar actions of cohomogeneity two.

We show that all vector bundles over CP^2 which are not spin admit a complete metric with nonnegative sectional curvature. In the proof we construct a nonnegatively curved metric on the corresponding principle bundle by showing that it admits a cohomogeneity one action with singular orbits of codimension 2. This is clo…

The study classifies manifolds with specific rational cohomology properties.

We give a characterization of those Alexandrov spaces admitting a cohomogeneity one action of a compact connected Lie group $G$ for which the action is Cohen--Macaulay. This generalizes a similar result for manifolds to the singular setting of Alexandrov spaces where, in contrast to the manifold case, we find several a…

In this paper we give a classification of closed and connected Lie groups, up to conjugacy in $Iso({\mathbb{R}^3_1})$, acting by cohomogeneity one on the three dimensional Minkowski space $\mathbb{R}^3_1$ in both cases, proper and nonproper actions. Then we determine causal characters of the orbits.

We study hyperpolar actions on reducible symmetric spaces of the compact type. Our main result is that an indecomposable hyperpolar action on a symmetric space of the compact type is orbit equivalent to a Hermann action or of cohomogeneity one.

New Einstein solvmanifolds created from non-flat Ricci solitons.

A cohomogeneity one manifold is a manifold with the action of a compact Lie group, whose quotient is one dimensional. Such manifolds are of interest in Riemannian geometry, in the context of nonnegative sectional curvature, as well as in other areas of geometry and in physics. In this paper we classify compact simply c…

Study cohomogeneity one expanding Ricci solitons on specific topologies.