Non-naturally reductive Einstein metrics on exceptional Lie groups

Chrysikos, Ioannis; Sakane, Yusuke

doi:10.1016/j.geomphys.2017.01.030

Cited by 18 publications

(17 citation statements)

References 11 publications

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“…We denote such a Lie group G by

G true(α_{i_{0}} true) \equiv G true(i_{0} true)

. Following the notation of ,

G \equiv G true(i_{0} true)

is said to belong to the class of type

I_{b} false(3 false)

I I_{b} false(3 false)

, or

I I I_{b} false(3 false)

if after deleting the black vertex, the Dynkin diagram splits into one, two, or three components, respectively. And the classification is given as follows (see )…”

Section: Generalized Flag Manifolds With Second Betti Number B2false(mentioning

confidence: 99%

“…Consider the homogeneous manifold

{normalG}_{2} false(2 false) / A_{1}

with the decomposition

m = {frakturt}_{0} \oplus {frakturp}_{1} \oplus {frakturp}_{2} \oplus {frakturp}_{3},

and Ad A 1 ‐invariant metrics on

{normalG}_{2} false(2 false) / A_{1}

defined by

false⟨, false⟩ = u_{0} {\cdot B |}_{t_{0}} + x_{1} {\cdot B |}_{p_{1}} + x_{2} \cdot B {{false|}_{p_{2}} + x_{3} \cdot B |}_{p_{3}},

where

u_{0}, x_{1}, x_{2}, x_{3} \in {double-struckR}^{+}

. By Lemma and Lemma (or Corollary 4.3 in ), the components of Ricci tensor with respect to the metric are as follows:

({, \begin{matrix} r_{T} & = \frac{1}{4} u_{0} (\frac{1}{6 x_{2}^{2}} + \frac{3}{4 x_{3}^{2}} + \frac{1}{12 x_{1}^{2}}), \\ r_{{frakturp}_{1}} & = - \frac{1}{8 x_{1}^{2}} (\frac{u_{0}}{12} + \frac{2 x_{2}}{3}) + \frac{1}{16} (\frac{x_{1}}{x_{2} x_{3}} - \frac{x_{3}}{x_{<...>}}) \end{matrix})

…”

Section: Einstein Metrics On Homogenous Manifoldsmentioning

confidence: 99%

“…, or (3) if after deleting the black vertex, the Dynkin diagram splits into one, two, or three components, respectively. And the classification is given as follows (see [8]) 7 (3), 7 (5), 8 (2) (3) 6 (3)…”

Section: Generalized Flag Manifolds With Second Betti Number ( ) =mentioning

confidence: 99%

“…3)where 0 , 1 , 2 , 3 ∈ ℝ + . By Lemma 2.1 and Lemma 3.1 (or Corollary 5.2 in[8]), the components of Ricci tensor with respect to the metric (4.3) are as follows:…”

mentioning

confidence: 99%

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New Einstein–Randers metrics on some homogeneous manifolds

Tan

2018

Mathematische Nachrichten

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In this article, we find several new non‐Riemannian Einstein–Randers metrics on some homogeneous manifolds arising from the exceptional Lie groups. Based on the discussions about generalized flag manifolds with second Betti number b2false(Mfalse)=1, we firstly prove that there exists Riemannian Einstein metrics on these homogeneous manifolds. Then, we prove that there exists non‐Riemannian Einstein–Randers metrics on these homogeneous manifolds.

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“…We denote such a Lie group G by

G true(α_{i_{0}} true) \equiv G true(i_{0} true)

. Following the notation of ,

G \equiv G true(i_{0} true)

is said to belong to the class of type

I_{b} false(3 false)

I I_{b} false(3 false)

, or

I I I_{b} false(3 false)

if after deleting the black vertex, the Dynkin diagram splits into one, two, or three components, respectively. And the classification is given as follows (see )…”

Section: Generalized Flag Manifolds With Second Betti Number B2false(mentioning

confidence: 99%

“…Consider the homogeneous manifold

{normalG}_{2} false(2 false) / A_{1}

with the decomposition

m = {frakturt}_{0} \oplus {frakturp}_{1} \oplus {frakturp}_{2} \oplus {frakturp}_{3},

and Ad A 1 ‐invariant metrics on

{normalG}_{2} false(2 false) / A_{1}

defined by

false⟨, false⟩ = u_{0} {\cdot B |}_{t_{0}} + x_{1} {\cdot B |}_{p_{1}} + x_{2} \cdot B {{false|}_{p_{2}} + x_{3} \cdot B |}_{p_{3}},

where

u_{0}, x_{1}, x_{2}, x_{3} \in {double-struckR}^{+}

. By Lemma and Lemma (or Corollary 4.3 in ), the components of Ricci tensor with respect to the metric are as follows:

({, \begin{matrix} r_{T} & = \frac{1}{4} u_{0} (\frac{1}{6 x_{2}^{2}} + \frac{3}{4 x_{3}^{2}} + \frac{1}{12 x_{1}^{2}}), \\ r_{{frakturp}_{1}} & = - \frac{1}{8 x_{1}^{2}} (\frac{u_{0}}{12} + \frac{2 x_{2}}{3}) + \frac{1}{16} (\frac{x_{1}}{x_{2} x_{3}} - \frac{x_{3}}{x_{<...>}}) \end{matrix})

…”

Section: Einstein Metrics On Homogenous Manifoldsmentioning

confidence: 99%

Section: Generalized Flag Manifolds With Second Betti Number ( ) =mentioning

confidence: 99%

“…3)where 0 , 1 , 2 , 3 ∈ ℝ + . By Lemma 2.1 and Lemma 3.1 (or Corollary 5.2 in[8]), the components of Ricci tensor with respect to the metric (4.3) are as follows:…”

mentioning

confidence: 99%

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New Einstein–Randers metrics on some homogeneous manifolds

Tan

2018

Mathematische Nachrichten

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“…In 2014, by using the methods of representation theory, Chen and Liang [11] found a non-naturally reductive Einstein metric on the compact simple Lie group F 4 . More recently, Chrysikos and Sakane proved that there exists non-naturally reductive Einstein metric on exceptional Lie groups, espeacially for G 2 , they gave the first example of non-naturally reductive Einstein metric (see [13]).…”

Section: Introductionmentioning

confidence: 99%

New non-naturally reductive Einstein metrics on exceptional simple Lie groups

Chen

Deng

2018

Journal of Geometry and Physics

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In this article, we achieved several non-naturally reductive Einstein metrics on exceptional simple Lie groups, which are formed by the decomposition arising from general Wallach spaces. By using the decomposition corresponding to the two involutive automorphisms, we calculated the non-zero coefficients in the expression for the components of Ricci tensor with respect to the given metrics. The Einstein metrics are obtained as solutions of systems polynomial equations, which we manipulate by symbolic computations using Gröbner bases.

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On Left-Invariant Einstein Riemannian Metrics That Are Not Geodesic Orbit

Nikonorov

2018

Transformation Groups

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In this paper we prove that the compact Lie group G2 admits a left-invariant Einstein metric that is not geodesic orbit. In order to prove the required assertion, we develop some special tools for geodesic orbit Riemannian manifolds. It should be noted that a suitable metric is discovered in a recent paper by I. Chrysikos and Y. Sakane, where the authors proved also that this metric is not naturally reductive.2010 Mathematical Subject Classification: 53C20, 53C25, 53C35.

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Non-naturally reductive Einstein metrics on exceptional Lie groups

Cited by 18 publications

References 11 publications

New Einstein–Randers metrics on some homogeneous manifolds

New Einstein–Randers metrics on some homogeneous manifolds

New non-naturally reductive Einstein metrics on exceptional simple Lie groups

On Left-Invariant Einstein Riemannian Metrics That Are Not Geodesic Orbit

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