Paper · Feb 18, 2024
On properties of the sets of positively curved Riemannian metrics on generalized Wallach spaces
As on arXiv
Abstract
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 the Wallach spaces , and which admit positive sectional curvature and belong to a given invariant surface of the normalized Ricci flow we established that they form a set bounded by three connected and pairwise disjoint regular space curves such that each of them approaches two others asymptotically at infinity. Analogously, for all generalized Wallach spaces the set of Riemannian metrics which belong to and admit positive Ricci curvature is bounded by three curves each consisting of two connected components as regular curves. Intersections and asymptotical behaviors of these components were studied as well.
Comments: 3 figures
Its neighbourhood
Connected papers
The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2021, plus the canon.
-
On the dynamics of a three-dimensional differential system related to the normalized Ricci flow on generalized Wallach spaces
Topics Sphere theorems · Same author Nurlan Abiev · Shared terms “Wallach spaces”, “generalized Wallach”, “normalized Ricci”
-
Ricci curvature and normalized Ricci flow on generalized Wallach spaces
Topics Ricci flow · Same author Nurlan Abiev · Shared terms “generalized Wallach”, “Wallach spaces”, “normalized Ricci”
-
Positive sectional curvature is not preserved under the Ricci flow in dimensions seven and thirteen
Topics Sphere theorems · Shared terms “Wallach spaces”, “positively curved”, “positive sectional”
-
The complete dynamics description of positively curved metrics in the Wallach flag manifold
Topics Sphere theorems · Shared terms “Wallach”, “positively curved”, “invariant Riemannian”
-
Ricci Flow Preserves Positive Sectional Curvature on Homogeneous Spheres
Topics Sphere theorems · Shared terms “positively curved”, “positive sectional”, “sectional curvature”
-
The Ricci curvature and the normalized Ricci flow on the Stiefel manifolds
Topics Ricci flow · Same author Nurlan Abiev · Shared terms “normalized Ricci”, “Riemannian metrics”, “positive Ricci”
-
Refined behavior description of the normalized Ricci flow on homogeneous spaces
Topics Ricci flow · Same author Nurlan Abiev · Shared terms “generalized Wallach”, “Wallach spaces”, “normalized Ricci”
-
Rigidity of area non-increasing maps
Topics Sphere theorems · Shared terms “positively curved”
Read around it
Background reading
The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.
In the guidebook
- 11A.11Hamilton’s 1982 Theorem
Positive Ricci curvature in dimension 3 flows to a round metric.
- 5A.5Uniformization and the Two-Dimensional Ricci Flow
Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.
- 10A.12Curve Shortening and Minimal Discs
How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.