The classification of homogeneous Einstein metrics on flag
manifolds with b2(M)=1

J 2013

The classification of homogeneous Einstein metrics on flag manifolds with b2(M)=1

CHRYSIKOS, Ioannis a Yusuke SAKANE

Základní údaje

Originální název

The classification of homogeneous Einstein metrics on flag manifolds with b2(M)=1

Autoři

CHRYSIKOS, Ioannis a Yusuke SAKANE

Vydání

Bulletin des Sciences Mathématiques, 2013, 0007-4497

Další údaje

Jazyk

angličtina

Typ výsledku

Článek v odborném periodiku

Obor

10101 Pure mathematics

Stát vydavatele

Dánsko

Utajení

není předmětem státního či obchodního tajemství

Odkazy

URL

Impakt faktor

Impact factor: 0.733

Označené pro přenos do RIV

Ano

Kód RIV

RIV/00216224:14310/13:00067050

Organizační jednotka

Přírodovědecká fakulta

DOI

https://doi.org/10.1016/j.bulsci.2013.11.002

UT WoS

000342250700001

Klíčová slova anglicky

Homogeneous Einstein metric; Flag manifold; Second Betti number; Riemannian submersion; Finiteness conjecture; Twistor fibration

Štítky

AKR

Změněno: 4. 4. 2014 11:04, doc. Dr. Ioannis Chrysikos

Anotace

V originále

Consider a compact connected simple Lie group G. We study homogeneous Einstein metrics for a class of compact homogeneous spaces, namely generalized flag manifolds G/H with second Betti number b2(G/H)=1. There are 8 infinite families G/H corresponding to a classical simple Lie group G and 25 exceptional flag manifolds, which all have some common geometric features; for example they admit a unique invariant complex structure which gives rise to unique invariant Kähler–Einstein metric. The most typical examples are the compact isotropy irreducible Hermitian symmetric spaces for which the Killing form is the unique homogeneous Einstein metric (which is Kähler). For non-isotropy irreducible spaces the classification of homogeneous Einstein metrics has been recently completed for 24 of the 26 cases. In this paper we construct the Einstein equation for the two unexamined cases, namely the flag manifolds E8/U(1)×SU(4)×SU(5) and E8/U(1)×SU(2)×SU(3)×SU(5). In order to determine explicitly the Ricci tensors of an E8-invariant metric we use a method based on the Riemannian submersions. For both spaces we classify all homogeneous Einstein metrics and thus we conclude that any flag manifold G/H with b2(M)=1 admits a finite number of non-isometric non-Kähler invariant Einstein metrics. The precise number of these metrics is given in Table 1.

Návaznosti

GBP201/12/G028, projekt VaV

Název: Ústav Eduarda Čecha pro algebru, geometrii a matematickou fyziku

Investor: Grantová agentura ČR, Ústav Eduarda Čecha pro algebru, geometrii a matematickou fyziku

Citovat

CHRYSIKOS, Ioannis a Yusuke SAKANE. The classification of homogeneous Einstein metrics on flag manifolds with b2(M)=1. Bulletin des Sciences Mathématiques. 2013, roč. 138/2014, in press, s. in press, in press. ISSN 0007-4497. Dostupné z: https://doi.org/10.1016/j.bulsci.2013.11.002.

@article{1174158,
   author = {Chrysikos, Ioannis and Sakane, Yusuke},
   article_number = {in press},
   doi = {https://doi.org/10.1016/j.bulsci.2013.11.002},
   keywords = {Homogeneous Einstein metric; Flag manifold; Second Betti number; Riemannian submersion; Finiteness conjecture; Twistor fibration},
   language = {eng},
   issn = {0007-4497},
   journal = {Bulletin des Sciences Mathématiques},
   title = {The classification of homogeneous Einstein metrics on flag manifolds with b2(M)=1},
   url = {http://dx.doi.org/10.1016/j.bulsci.2013.11.002},
   volume = {138/2014},
   year = {2013}
}

TY  - JOUR
ID  - 1174158
AU  - Chrysikos, Ioannis - Sakane, Yusuke
PY  - 2013
TI  - The classification of homogeneous Einstein metrics on flag manifolds with b2(M)=1
JF  - Bulletin des Sciences Mathématiques
VL  - 138/2014
IS  - in press
SP  - in press
EP  - in press
SN  - 00074497
KW  - Homogeneous Einstein metric
KW  - Flag manifold
KW  - Second Betti number
KW  - Riemannian submersion
KW  - Finiteness conjecture
KW  - Twistor fibration
UR  - http://dx.doi.org/10.1016/j.bulsci.2013.11.002
L2  - http://dx.doi.org/10.1016/j.bulsci.2013.11.002
N2  - Consider a compact connected simple Lie group G. We study homogeneous Einstein metrics for a class of compact homogeneous spaces, namely generalized flag manifolds G/H with second Betti number b2(G/H)=1. There are 8 infinite families G/H corresponding to a classical simple Lie group G and 25 exceptional flag manifolds, which all have some common geometric features; for example they admit a unique invariant complex structure which gives rise to unique invariant Kähler–Einstein metric. The most typical examples are the compact isotropy irreducible Hermitian symmetric spaces for which the Killing form is the unique homogeneous Einstein metric (which is Kähler). For non-isotropy irreducible spaces the classification of homogeneous Einstein metrics has been recently completed for 24 of the 26 cases. In this paper we construct the Einstein equation for the two unexamined cases, namely the flag manifolds E8/U(1)×SU(4)×SU(5) and E8/U(1)×SU(2)×SU(3)×SU(5). In order to determine explicitly the Ricci tensors of an E8-invariant metric we use a method based on the Riemannian submersions. For both spaces we classify all homogeneous Einstein metrics and thus we conclude that any flag manifold G/H with b2(M)=1 admits a finite number of non-isometric non-Kähler invariant Einstein metrics. The precise number of these metrics is given in Table 1.
ER  -

CHRYSIKOS, Ioannis a Yusuke SAKANE. The classification of homogeneous Einstein metrics on flag manifolds with b2(M)=1. \textit{Bulletin des Sciences Mathématiques}. 2013, roč.~138/2014, in press, s.~in press, in press. ISSN~0007-4497. Dostupné z: https://doi.org/10.1016/j.bulsci.2013.11.002.

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