Manifolds Of Cohomogeneity Research Articles

Type I Almost-Homogeneous Manifolds of Cohomogeneity One—IV

This paper is one of a series in which we generalize our earlier results on the equivalence of existence of Calabi extremal metrics to the geodesic stability for any type I compact complex almost homogeneous manifolds of cohomogeneity one. In this paper, we actually carry all the earlier results to the type I cases. In Part II, we obtained a substantial amount of new Kähler–Einstein manifolds as well as Fano manifolds without Kähler–Einstein metrics. In particular, by applying Theorem 15 therein, we obtained complete results in the Theorems 3 and 4 in that paper. However, we only have partial results in Theorem 5. In this note, we provide a report of recent progress on the Fano manifolds N n , m when n > 15 and N n , m ′ when n > 4 . We provide two pictures for these two classes of manifolds. See Theorems 1 and 2 in the last section. Moreover, we present two conjectures. Once we solve these two conjectures, the question for these two classes of manifolds will be completely solved. By applying our results to the canonical circle bundles, we also obtain Sasakian manifolds with or without Sasakian–Einstein metrics. These also provide open Calabi–Yau manifolds.

Moser Vector Fields and Geometry of the Mabuchi Moduli Space of Kähler Metrics

There is a natural Moser type transformation along any curve in the moduli spaces of Kähler metrics. In this paper we apply this transformation to give an explicit construction of the parallel transformation along a curve in the Mabuchi moduli space of Kähler metrics. This is crucial in the proof of the equivalence between the existence of the Kähler metrics with constant scalar curvature and the geodesic stability for the type II compact almost homogeneous manifolds of cohomogeneity one mentioned in (Guan 2013). We also explain a new description of the geodesics and prove a curvature property of the moduli space, called curvature symmetric, which makes it similar to some special symmetric spaces with nonpositive curvatures, although the spaces are usually not complete. Finally, we generalize our geodesic stability conjectures in (Guan 2003) and give several results on the Lie algebra structures related to the parallel transformations. In the last section, we generalize the Futaki obstruction of the Kähler-Einstein metrics to the parallel vector fields of the invariant Mabuchi moduli space. We call the related stability the parallel stability. This includes the toric and cohomogeneity one cases as well as the spherical manifolds.

Six-Dimensional Nearly Kähler Manifolds of Cohomogeneity One (II)

Let M be a six dimensional manifold, endowed with a cohomogeneity one action of G = SU2 × SU2, and \({M_{\rm reg} \subset M}\) its subset of regular points. We show that Mreg admits a smooth, 2-parameter family of G-invariant, non-isometric strict nearly Kahler structures and that a 1-parameter subfamily of such structures smoothly extends over a singular orbit of type S3. This determines a new class of examples of nearly Kahler structures on T S3.

Type II almost-homogeneous manifolds of cohomogeneity one

Type II almost-homogeneous manifolds of cohomogeneity one

Spaces admitting homogeneous [formula omitted]-structures

We classify all seven-dimensional manifolds which admit a homogeneous cosymplectic G 2 -structure. The motivation for this classification is that each of these spaces is a possible principal orbit of a parallel Spin ( 7 ) -manifold of cohomogeneity one.

Six-dimensional nearly Kähler manifolds of cohomogeneity one

We consider six-dimensional strict nearly Kähler 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.

Affine compact almost-homogeneous manifolds of cohomogeneity one

AbstractThis paper is one in a series generalizing our results in [12, 14, 15, 20] on the existence of extremal metrics to the general almost-homogeneous manifolds of cohomogeneity one. In this paper, we consider the affine cases with hypersurface ends. In particular, we study the existence of Kähler-Einstein metrics on these manifolds and obtain new Kähler-Einstein manifolds as well as Fano manifolds without Kähler-Einstein metrics. As a consequence of our study, we also give a solution to the problem posted by Ahiezer on the nonhomogeneity of compact almost-homogeneous manifolds of cohomogeneity one; this clarifies the classification of these manifolds as complex manifolds. We also consider Fano properties of the affine compact manifolds.

SU(3)-manifolds of cohomogeneity one

In this article, we classify 7- and 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) \({M/G\cong S^{1}}\) and the principal orbits are simply connected. We discuss applications to the study of the group manifold SU(3) and to 8-dimensional quaternion-Kahler spaces, and links between dimension 7 and 8 given by circle actions.

Existence of extremal metrics on almost homogeneous manifolds of cohomogeneity one—II

We consider the existence of Calabi extremal metrics on certain compact almost homogeneous manifold of cohomogeneity one. We proved that the positivity of the generalized Futaki invariant implies the existence. We expect that the converse is also true for all this kind of manifolds.

QUATERNIONIC KAHLER MANIFOLDS OF COHOMOGENEITY ONE

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 cohomogeneity one with respect to a group of 3-Sasakian symmetries. Information is also obtained about non-compact quaternionic Kähler manifolds of cohomogeneity one and the cohomogeneity of adjoint orbits in complex semi-simple Lie algebras.