Geometry of homogeneous Riemannian manifolds

The evolution of positively curved invariant Riemannian metrics on the Wallach spaces under the Ricci flow

“…on one special class of Riemannian manifolds M n called generalized Wallach spaces (or three-locally-symmetric spaces in other terms) according to the definitions of [22,26], where g(t) means a 1-parameter family of Riemannian metrics, Ric g is the Ricci tensor and S g is the scalar curvature of the Riemannian metric g. Generalized Wallach spaces are characterized as compact homogeneous spaces G/H whose isotropy representation decomposes into a direct sum p = p 1 ⊕ p 2 ⊕ p 3 of three Ad(H )-invariant irreducible modules satisfying [22,24]. The complete classification of generalized Wallach spaces is obtained recently (independently) in the papers [15,25].…”

On half conformally flat four-dimensional Lie algebras

“…It follows from (6) and (7) that the components R ijkt of the curvature tensor, the components r sm of the Ricci tensor, the components W pqef of the Weyl tensor, the elements of the matrices , and the scalar cur vature s are functions of the structure constants and the components g ij of the metric tensor (see [2,6]). = Dα, = -= -D < 0.…”

Convex polyhedra in hyperbolic space and interpolation of functions

“…Conformally flat metrics of bounded curvature correspond to convex surfaces in hyperbolic space [1,2]. The convex sets most important in practice are convex polyhedra.…”

“…An expression of the form ds 2 = , where f ∈ C 2 (R n ) is a positive function, determines a conformally flat metric on R n with one dimensional curvature in the direction of a unit vector ξ (see [1,2,4]), which equals…”

“…The mapping Z f is an isometric embedding of {R n , ds 2 } into the isotropic cone C + (see [1,2]). Consider the mappings where κ > 0 is a fixed constant; we have…”