Revisiting homogeneous spaces with positive curvature

On the Geometry of Homogeneous Spaces of Positive Curvature

This Master’s thesis is a study of some of the geometric properties of Riemannian homogeneous spaces. These are Riemannian manifolds M equipped with a transitive group of isometries G, meaning that the local geometry of the manifold is the same at every point. In Section 1.3 we see that such spaces are diffeomorphic to the quotient G/K, where G is a Lie group and K is the isotropy group at the ...

On Riemannian Curvature of Homogeneous Spaces

Curvature homogeneous spaces whose curvature tensors have large symmetries

We study curvature homogeneous spaces or locally homogeneous spaces whose curvature tensors are invariant by the action of “large” Lie subalgebras h of so(n). In this paper we deal with the cases of h = so(r)⊕ so(n− r) (2 ≤ r ≤ n− r), so(n− 2), and the Lie algebras of Lie groups acting transitively on spheres, and classify such curvature homogeneous spaces or locally homogeneous spaces.

Geometrically Formal Homogeneous Metrics of Positive Curvature

A Riemannian manifold is called geometrically formal if the wedge product of harmonic forms is again harmonic, which implies in the compact case that the manifold is topologically formal in the sense of rational homotopy theory. A manifold admitting a Riemannian metric of positive sectional curvature is conjectured to be topologically formal. Nonetheless, we show that among the homogeneous Riem...

Quasi-positive Curvature on Homogeneous Bundles

We provide new examples of manifolds which admit a Riemannian metric with sectional curvature nonnegative, and strictly positive at one point. Our examples include the unit tangent bundles of CP, HP and CaP, and a family of lens space bundles over CP.