Invariant Einstein metrics on Ledger–Obata spaces

We study invariant metrics on Ledger-Obata spaces F m / diag(F ). We give the classification and an explicit construction of all naturally reductive metrics, and also show that in the case m = 3, any invariant metric is naturally reductive. We prove that a Ledger-Obata space is a geodesic orbit space if and only if the metric is naturally reductive. We then show that a Ledger-Obata space is reducible if and only if it is isometric to the product of Ledger-Obata spaces (and give an effective method of recognising reducible metrics), and that the full connected isometry group of an irreducible Ledger-Obata space F m / diag(F ) is F m . We deduce that a Ledger-Obata space is a geodesic orbit manifold if and only if it is the product of naturally reductive Ledger-Obata spaces.2010 Mathematics Subject Classification. 53C30, 53C25, 17B20.

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“…. Then by Lemma 2(iii) we have (5). An adapted system is called super-adapted if in addition it satisfies (7).…”

Section: Geodesic Orbit Metrics On Ledger-obata Spacesmentioning

confidence: 97%

“…, s, with the same Einstein constants, then ρ is also Einstein. Such Einstein metrics are called routine in [5]. It is known that all Einstein invariant metrics on Ledger-Obata spaces are routine for m ≤ 4.…”

Section: Proposition 1 ([6 Proposition 3]) Every Invariant Riemannimentioning

confidence: 99%

On invariant Riemannian metrics on Ledger–Obata spaces

Nikolayevsky

Nikonorov

2018

manuscripta math.

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“…The only other Einstein metric on H 3 / diag(H) is the normal homogeneous metric induced by Q; see [15]. It corresponds to the origin in Figure 7 and is a strict local maximum of the associated functional S |M T .…”

Section: Under the Constraint Trmentioning

confidence: 99%

On the variational properties of the prescribed Ricci curvature functional

Pulemotov¹,

Ziller²

2021

Preprint

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We study the prescribed Ricci curvature problem for homogeneous metrics. Given a (0,2)-tensor field T , this problem asks for solutions to the equation Ric(g) = cT for some constant c. Our approach is based on examining global properties of the scalar curvature functional whose critical points are solutions to this equation. We produce conditions under which it has a global maximum and find a large of class of spaces where it admits saddle critical points, even though the Palais-Smale condition fails to hold. Finally, we consider several examples, including ones where the functional admits critical submanifolds of various dimensions.

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“…Note that for k = 2 we get irreducible symmetric spaces. For k = 3 and k = 4, the classification of invariant Einstein metrics on the spaces F k / diag(F ) was obtained in [15] and [9] respectively, but for k ≥ 5 this problem is open.…”

Section: Description Of Einstein Metrics On Generalized Wallach Spacesmentioning

confidence: 99%

“…In particular, it is proved that there are at most four Einstein metrics (up to homothety) for every such space. In [9], the authors classified all invariant Einstein metrics on Ledger-Obata spaces F 4 / diag(F ), hence, Einstein metrics of generalized Wallach spaces in the item 3) of Theorem 1. By the results from [8], [13] and [17], all invariant Einstein metrics on generalized Wallach spaces in the item 2) of Theorem 1 except SO(k + l + m)/SO(k) × SO(l) × SO(m) were classified.…”

Section: Introductionmentioning

confidence: 99%

Invariant Einstein metrics on generalized Wallach spaces

Chen

Nikonorov

2018

Sci. China Math.

Self Cite

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Invariant Einstein metrics on generalized Wallach spaces have been classified except SO(k + l + m)/SO(k) × SO(l) × SO(m). In this paper, we give a survey on the study of invariant Einstein metrics on generalized Wallach spaces, and prove that there are infinitely many spaces of the type SO(k + l + m)/SO(k) × SO(l) × SO(m) admitting exactly two, three, or four invariant Einstein metrics up to a homothety.

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Invariant Einstein metrics on Ledger–Obata spaces

Cited by 8 publications

References 18 publications

On invariant Riemannian metrics on Ledger–Obata spaces

On invariant Riemannian metrics on Ledger–Obata spaces

On the variational properties of the prescribed Ricci curvature functional

Invariant Einstein metrics on generalized Wallach spaces

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