A real variety with boundary and its global parameterization

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

“…. The function β(t) is well defined, continuous and positive valued for t ∈ (0, m) or t ∈ (M, +∞), where m and M are given in (11), 0 < m < M . It follows then the components r ki and r kj of r k are respectively continuous images of the connected sets (0, m) and (M, +∞) under a vector-function with coordinates x i (t), x j (t) and x k (t).…”

“…Since then studies related to this topic were continued in [1,3,7,22] concerning classifications of singular (equilibria) points of (4) being Einstein metrics and their bifurcations. The authors of [2,10,11] studied an interesting and quite complicated surface of bifurcations of (4) defined by a symmetric polynomial equation in three variables a 1 , a 2 , a 3 of degree 12. In the sequel authors of [4,8] considered the evolution of positively curved Riemannian metrics under the influence of ( 5) on an interesting class of generalized Wallach spaces with coincided parameters a 1 = a 2 = a 3 := a ∈ (0, 1/2) generalizing some results of [14,15].…”

Invariant Einstein metrics on generalized Wallach spaces

“…The part of Ω in (0, 1/2) 3 consists of three (pairwise isometric) "bubbles" spanned on every pair of "edges". Note also that in [5], the author found an explicit parameterization of the surface Ω. Another important fact is that the set (0, 1/2) 3 \ Ω has exactly three connected components.…”

“…Direct calculations (see see (2)) show that ∂Q ∂a 1 − ∂Q ∂a 2 = (a 2 − a 1 ) ∂Q ∂s 2 + a 3 ∂Q ∂s 3 = 8(a 1 − a 2 ) · Q 12 , where 1 a 3 2 a 2 3 + 12288a 5 1 a 2 2 a 3 3 + 4096a 5…”

On Degenerated Singular Points of Dynamical Systems Related to the Normalized Ricci Flow on Generalized Wallach Spaces

“…s 1 = a 1 + a 2 + a 3 , s 2 = a 1 a 2 + a 1 a 3 + a 2 a 3 и s 3 = a 1 a 2 a 3 . Свойства алгебраической поверхности Ω, определяемой уравнением Q(a 1 , a 2 , a 3 ) = 0, изучены в работах [8][9][10]. Поставим теперь вопрос о том, при каких (a 1 , a 2 , a 3 ) ∈ R 3 система (4) может допускать хотя бы одну особую точку, удовлетворяющую усло-О вырожденных особых точках динамических систем вию σ = δ = 0.…”