arXiv research

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…

New proof for curved 3-cohom manifold rational ellipticity.

Constructs harmonic maps between special geometric shapes.

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…

We prove a structure theorem for closed topological manifolds of cohomogeneity one; this result corrects an oversight in the literature. We complete the equivariant classification of closed, simply connected cohomogeneity one topological manifolds in dimensions $5$, $6$, and $7$ and obtain topological characterizations…

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

Constructs biharmonic and $r$-harmonic submanifolds in cohomogeneity one manifolds.

Classifies manifolds with quasipositive curvature.

We show that a certain family of cohomogeneity one manifolds does not admit an invariant metric of nonnegative sectional curvature, unless it admits one with positive curvature. As a consequence, the classification of nonnegatively curved cohomogeneity one manifolds in dimension 7 is reduced to only one further family …

Abstract: Characterizes spaces with positive scalar curvature.

Study $G_2$ manifolds with symmetry, finding new submanifolds.

We construct several infinite families of nonnegatively curved manifolds of low cohomogeneity and small dimension which can be distinguished by their cohomology rings. In particular, we exhibit an infinite family of eight-dimensional cohomogeneity one manifolds of nonnegative curvature with pairwise non-isomorphic comp…

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

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

In this paper we give a full diffeomorphism characterization of compact simply connected cohomogeneity one manifolds in dimension six.

Classifies Einstein metrics on 4-manifolds with specific symmetry groups.

Study geometric equations on cohomogeneity one manifolds near singular orbits.

We classify simply connected, closed cohomogeneity one manifolds with singly generated or 4-periodic rational cohomology and positive Euler characteristic.

We classify those manifolds of positive euler characteristic on which a lie group G acts with cohomogeneity one, where G is classical simple

The study classifies manifolds with specific rational cohomology properties.

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…

Solves initial value problem for harmonic maps on specific manifolds.

In this paper we give a characterization of the possible homology groups that can occur for compact simply connected cohomogeneity one manifolds in dimensions seven and lower.

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…

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

Study on special Hermitian metrics on cohomogeneity one manifolds.

We consider 6-dimensional strict nearly Kaehler manifolds acted on by a compact, cohomogeneity one automorphism group G. We classify the compact manifolds of this class up to G-diffeomorphisms. We also prove that the manifold has constant sectional curvature whenever the group G is simple.

Paper proves non-existence of certain balanced metrics on six-manifolds.

We give an exhaustive description of all simply connected odd dimensional cohomogeneity one manifolds that can possibly support an invariant metric with positive sectional curvature. Among the known examples of odd dimensional manifolds with positive curvature, apart from spheres, there are two infinite families among …

No Einstein metrics found on certain double disk bundles.

We classify SO(n)-equivariant principal bundles over $S^n$ in terms of their isotropy representations over the north and south poles. This is an example of a general result classifying equivariant $(Π, G)$-bundles over cohomogeneity one manifolds.

Study GKM actions on special manifolds with interval orbit spaces.

We analyse some properties of the cohomogeneity one Ricci soliton equations, and use Ansatze of cohomogeneity one type to produce new explicit examples of complete Kahler Ricci solitons of expanding, steady and shrinking types. These solitons are foliated by hypersurfaces which are circle bundles over a product of Fano…

New steady gradient Ricci solitons found on specific four-manifolds.

This is a survey on cohomogeneity one manifolds with positive curvature. We discuss the known examples of this type and their geometry and the functions that describe the metric. We also describe the classification of cohomogeneity one manifolds that can admit a metric with positive curvature due to Grove-Wilking-Zille…

Researchers classify curvature homogeneous metrics on 4D manifolds.

The study finds infinite nodal solutions for equations on positive Ricci curvature manifolds.

We show that in cohomogeneity 3 there are G-manifolds with any given number of isolated singular orbits and an invariant metric of positive Ricci curvature. We show that the corresponding result is also true in cohomogeneity 5 provided the number of singular orbits is even.

We consider hypersurfaces in Einstein-Sasaki 5-manifolds which are tangent to the characteristic vector field. We introduce evolution equations that can be used to reconstruct the 5-dimensional metric from such a hypersurface, analogous to the (nearly) hypo and half-flat evolution equations in higher dimensions. We use…

In this paper, we classify compact simply connected cohomogeneity one manifolds up to equivariant diffeomorphism whose isotropy representation by the connected component of the principal isotropy subgroup has three or less irreducible summands. The manifold is either a bundle over a homogeneous space or an irreducible …

Study on metrics of non-negative curvature on vector bundles over specific manifolds.

Study cohomogeneity one solitons for $G_2$-structures on various manifolds.

Study shows deformations of quaternionic Kähler manifolds are locally inhomogeneous.

Study complete cohomogeneity one hypersurfaces in hyperbolic space.

The paper studies quaternionic Kähler manifolds and their fibration by solvmanifolds.

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 …

In this paper, we classify 8-dimensional manifolds M admitting an SU(3) action of cohomogeneity one such that (i) M is simply connected and the orbit space M/G is isomorphic to [0,1], and (ii) M/G=S^1 and the principal orbits are simply connected. We discuss applications to the study of the group manifold SU(3) and to …

Study on symmetries of quaternionic Kähler manifolds with S^1-symmetry.