in this paper, we give a classification of non simply connected seven dimensional reimannian manifolds of constant positive curvature which admit irreducible cohomogeneity-one actions. we characterize the acting groups and describe the orbits. the first and second homo-topy groups of the orbits have been presented as well.

Non-negatively Curved Cohomogeneity One Manifolds Chenxu He Prof. Wolfgang Ziller, Advisor A Riemannian manifold M is called cohomogeneity one if it admits an isometric action by a compact Lie group G and the orbit space is one dimension. Many new examples of non-negatively curved manifolds were discovered recently in this category. However not every cohomogeneity one manifold carries an invari...

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 main result in [12] to show that there are no noncompact cohomogeneity one s...

A manifold with a smooth action of a Lie group G is called G-manifold. In this paper we consider a complete Riemannian manifold M with the action of a closed and connected Lie subgroup G of the isometries. The dimension of the orbit space is called the cohomogeneity of the action. Manifolds having actions of cohomogeneity zero are called homogeneous. A classic theorem about Riemannian manifolds...

We consider compact manifolds $M$ with a cohomogeneity one action of Lie group $G$ such that the orbit space $M/G$ is closed interval. For $T$ maximal torus $G$, we find necessary and sufficient conditions on diagram $T$-action GKM type, describe its graph. The general results are illustrated explicit examples.

We characterize cohomogeneity one manifolds and homogeneous spaces with a compact Lie group action admitting an invariant metric positive scalar curvature.

There are very few known examples of manifolds with positive sectional curvature. Apart from the compact rank one symmetric spaces, they exist only in dimensions 24 and below and are all obtained as quotients of a compact Lie group equipped with a biinvariant metric under an isometric group action. They consist of certain homogeneous spaces in dimensions 6, 7, 12, 13 and 24 due to Berger [Be], ...