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Sets related to positively curved invariant Riemannian metrics on generalized Wallach spaces are considered. The problem arises in studying of the evolution of such metrics under the normalized Ricci flow equation. For Riemannian metrics of…

This paper is devoted to the study of the evolution of positively curved metrics on the Wallach spaces $SU(3)/T_{\max}$, $Sp(3)/Sp(1)\times Sp(1)\times Sp(1)$, and $F_4/Spin(8)$. We prove that for all Wallach spaces, the normalized Ricci…

We consider the asymptotic behavior of the normalized Ricci flow on generalized Wallach spaces that could be considered as special planar dynamical systems. All non symmetric generalized Wallach spaces can be naturally parametrized by three…

We proved that the normalized Ricci flow does not preserve the positivity of Ricci curvature of Riemannian metrics on every generalized Wallach space with $a_1+a_2+a_3\le 1/2$, in particular on the spaces…

We study topological structures of the sets $(0,1/2)^3 \cap \Omega$ and $(0,1/2)^3 \setminus \Omega$, where~$\Omega$ is one special algebraic surface defined by a symmetric polynomial in variables $a_1,a_2,a_3$ of degree~$12$. These…

We give a global picture of the Ricci flow on the space of three-dimensional, unimodular, nonabelian metric Lie algebras considered up to isometry and scaling. The Ricci flow is viewed as a two-dimensional dynamical system for the evolution…

We study the behavior of the normalized Ricci flow of invariant Riemannian homogeneous metrics at infinity for generalized Wallach spaces, generalized flag manifolds with four isotropy summands and second Betti number equal to one, and the…

We proved that on every Stiefel manifold $V_2\mathbb{R}^n\cong \operatorname{SO}(n)/\operatorname{SO}(n-2)$ with $n\ge 3$ the normalized Ricci flow preserves the positivity of the Ricci curvature of invariant Riemannian metrics with…

This article deals with the problems of preserving the Ricci curvature positivity on homogeneous spaces under the normalized Ricci flow (NRF). We found out infinitely many generalized Wallach spaces (GWS) on which the positivity of the…

In this paper we investigate a kind of generalized Ricci flow which possesses a gradient form. We study the monotonicity of the given function under the generalized Ricci flow and prove that the related system of partial differential…

In this note, we show that the classical Wallach manifold $\mathrm{SU}(3)/\mathrm{T}^2$-admits metrics of positive intermediate Ricci curvature $(\mathrm{Ric}_d >0)$ for $d = 1, 2, 3, 4, 5$ that lose these properties under the homogeneous…

We prove that there exist $\mathsf{SU}_{3}$-invariant metrics on Aloff-Wallach spaces $W^7_{k_1, k_2}$, as well as $\mathsf{SU}_{5}$-invariant metrics on the Berger space $B^{13}$, which have positive sectional curvature and evolve under…

We give the global picture of the normalized Ricci flow on generalized flag manifolds with two or three isotropy summands. The normalized Ricci flow for these spaces descents to a parameter depending system of two or three ordinary…

Motivated by M\"uller-Haslhofer results on the dynamical stability and instability of Ricci-flat metrics under the Ricci flow, we obtain dynamical stability and instability results for pairs of Ricci-flat metrics and vanishing 3-forms under…

The family of invariant Riemannian manifolds in the Wallach flag manifold $\mathrm{SU}(3)/\mathrm{T}^2$ is described by three parameters $(x,y,z)$ of positive real numbers. By restricting such a family of metrics in the \emph{tetrahedron}…

In this paper we study the evolution of almost non-negatively curved (possibly singular) three dimensional metric spaces by Ricci flow. The non-negatively curved metric spaces which we consider arise as limits of smooth Riemannian manifolds…

The Ricci flow is a parabolic evolution equation in the space of Riemannian metrics of a smooth manifold. To some extent, Einstein equations give rise to a similar hyperbolic evolution. The present text is an introductory exposition to…

This book gives an introduction to fundamental aspects of generalized Riemannian, complex, and K\"ahler geometry. This leads to an extension of the classical Einstein-Hilbert action, which yields natural extensions of Einstein and…

In this work we illustrate some well-known facts about the evolution of $S^3$ under the Ricci flow. The Dirac flow we introduce allows us to describe the 4- dimensional metrics with constant curvature. Another new flow leads to the…

We review the main aspects of Ricci flows as they arise in physics and mathematics. In field theory they describe the renormalization group equations of the target space metric of two dimensional sigma models to lowest order in the…