Basic Properties of the Normalized Ricci Flow on Reductive Homogeneous Spaces
DOI:
https://doi.org/10.58649/fvcy5q82Keywords:
Ricci flow, Homogeneous space, Einstein metric, Dynamical system, First integral, Singular pointAbstract
In this paper we present tools for reducing the normalized Ricci flow equation to a system of nonlinear autonomous differential equations on reductive homogeneous spaces. Methods for equivalent transforming them into systems with polynomial right-hand sides are also discussed. In general case we also consider the interpretation of the geometric concepts from the perspective of the theory of dynamical systems, specifically, the one-to-one correspondence between the invariant Riemannian metrics of a given reductive homogeneous space and the singular (equilibrium) points of the resulting dynamical system, and the property of the metric volume function serving as a first integral of that dynamical system.
References
N. Abiev, On the dynamics of a three-dimensional differential system related to the normalized Ricci flow on generalized Wallach spaces, Results Math., 79(5) (2024), 19–33.
N.A. Abiev, The Ricci curvature and the normalized Ricci flow on generalized Wallach spaces, Math. Phys. Anal. Geom., 28(2) (2025), 11–35.
N.A. Abiev, A. Arvanitoyeorgos, Yu.G. Nikonorov, P. Siasos, The dynamics of the Ricci flow on generalized Wallach spaces, Differ. Geom. Appl., 35(Suppl.) (2014), 26–43.
R.H. Bamler, Recent developments in Ricci flows, Notices Am. Math. Soc., 68 (2021), 1486–1498.
V.N. Berestovskiĭ, Yu.G. Nikonorov, Riemannian manifolds and homogeneous geodesics, Springer Monographs in Mathematics, Cham: Springer, (2020).
R.S. Hamilton, Three-manifolds with positive Ricci curvature, J. Diff. Geom., 17 (1982), 255–306.
Yu.G. Nikonorov, On a class of homogeneous compact Einstein manifolds, Sib. Math. J., 41(1) (2000), 168–172.
Yu.G. Nikonorov, Classification of generalized Wallach spaces, Geom. Dedicata, 181(1) (2016), 193–212; correction: Geom. Dedicata, 214(1) (2021), 849–851.
J-S. Park, Y. Sakane, Invariant Einstein metrics on certain homogeneous spaces, Tokyo J. Math., 20(1) (1997), 51–61.
M. Statha, Ricci flow on certain homogeneous spaces, Ann. Glob. Anal. Geom., 62(1) (2022), 93–127.
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No datasets were generated or analysed during the current study.
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