Serviços Personalizados
Journal
Artigo
Indicadores
-
Citado por SciELO
Links relacionados
-
Similares em SciELO
Compartilhar
Revista de la Unión Matemática Argentina
versão impressa ISSN 0041-6932versão On-line ISSN 1669-9637
Rev. Unión Mat. Argent. v.47 n.2 Bahía Blanca jul./dez. 2006
Einstein metrics on flag manifolds
Evandro C. F. dos Santos* and Caio J. C. Negreiros**
*Partially supported by FUNCAP
**Partialy supported by CNPq grant 303695/2005-6 and Fapesp grant 02/10246-2
Abstract: In this survey we describe new invariant Einstein metrics on flag manifolds. Following closely San Martin-Negreiros's paper [26] we state results relating Kähler, (1,2)-symplectic and Einstein structures on flags. For the proofs see [11] and [10].
2000 Mathematics Subject Classification. 53C55; 58D17; 53C25; 22F30.
Key words and phrases. Flag manifolds, Einstein metrics, Semi-simple Lie groups.
We recall that a Riemannian metric on a manifold
is called Einstein if
for some constant
. As we know Einstein metrics form a special class of metrics on a given manifold (see [4]). In this note we announce properties of these metrics and new examples of Einstein metrics on flag manifolds as described in [11] and [10].
With this purpose in mind, we consider as being a complex semi-simple Lie algebra and
a simple root system for
. If
is an arbitrary subset of
,
denotes the roots spanned by
. We have
| (1) |
where is a Cartan subalgebra of
and
is the root space associated to the root
. Let
| (2) |
the canonical parabolic subalgebra determined by . Hence
| (3) |
is called a flag manifold, where
has the Lie algebra
and
is the normalizer of
in
. Each manifold
has a very rich complex geometry, containing families of invariant Hermitian structures denoted by
.
The case for
, i.e., the full flag manifold is nowadays well understood. Starting with the work of Borel (cf. [7]), the classification of all invariant Hermitian structures is known and it was derived in [26].
On the other hand, the case for
is much less known so far. Some partial results are derived in [27] and [28].
We now describe the contents of this survey. In the first two sections we discuss all the invariant Hermitian structures on and the associated Einstein system of equations. In Section 3 we present new invariant Einstein metrics on generalized flag manifolds of type
We suggest Besse's book [4] as a reference for Einstein manifolds.
In Section 4 we state the classification of all invariant Einstein metrics on and state some partial results relating Kähler, (1,2)-symplectic and Einstein structures on
.
For a very stimulating article see [1].
All manifolds and maps between them will be assumed to be in this survey.
We would like to thank to FAEP-UNICAMP and FAPESP (grant 02/10246-2) for the financial support.
1. General results on the invariant Hermitian geometry of flag manifolds
We denote by the Cartan-Killing form of
, and we fix a Weyl basis
for
. We define the compact real form of
, as the real subalgebra
|
where and
.
Let be the origin of
.
is identified with
where . Complexifying
we obtain
, which can be identified with
| (4) |
A -invariant almost complex structure
on
, is completely determined by a collection of numbers
,
.
A -invariant Riemannian metric
on
is completely characterized by the following inner product
on
| (5) |
where is definite-positive with respect to the Cartan-Killing form. On each irreducible component of
,
with
.
Consider
the conjugation of
relatively to
. Hence,
is a Hermitian form on
, that originates a
-invariant Hermitian form on
.
If denotes the corresponding Kähler form then
| (6) |
We recall that a almost-Hermitian manifold is said -symplectic if
when one of the vectors
,
,
is of type
, and the other two are of type
. If
is integrable and
, we say
is a Kähler manifold.
2. Ricci tensor and the Einstein system of equations
We now consider a
-orthogonal basis adapted to a decomposition of
In other words,
for some
and
if
with
Define, as in [29],
| (7) |
that is,
| (8) |
where in the second equation we take all indices with
Notice that
is independent of orthonormal frame chosen for
and
Furthermore, if is an element of Weyl´s group then
| (9) |
The following result is due to Wang-Ziller [29] (see also [2]):
Lemma 2.1. The components of the Ricci tensor of an
-invariant metric on
are given by:
| (10) |
where .
More generally, Arvanitoyeorgos proved in [3] the following result
Proposition 2.2. The Ricci tensor of an invariant metric on a flag manifold
is given by
We have the following non-homogeneous version of this equation
With each solution we associate the Einstein constant, which is defined as the value of the Ricci tensor when
is re-normalized to have unit volume.
Using the Einstein system of equations described above, we describe now the known and new Einstein metrics on as in [11] and [10].
a) The normal metric. We notice this metric is not Kähler.
b) Kähler-Einstein metrics
On the flag manifold (
, up to permutation there is a unique integrable structure
, and associated with it a unique (up to scaling) Kähler-Einstein metric (which corresponds to the choice
according to Matsushima [19] or [4]):
Thus, counting in the symmetry of this metric, we have Kähler-Einstein metrics on
.
c) The Arvanitoyeorgos metrics
Arvanitoyeorgos ([3]) considers for all metrics in
(
) satisfying
(
),
otherwise
The Einstein system is reduced to the equations in
whose solution is
and
. Counting permutations, we get
Arvanitoyeorgos metrics whose Einstein constant is seen to be
d) The Sakane-Senda metrics
Sakane and Senda in [25] consider metrics in (
) satisfying
(
or
),
otherwise
Again, the the Einstein system is reduced to two equations in
whose solution is
and
.
e) A new family
If we find another solution in
, for
and
.
f) Two new families
On (
) we consider
(
or
),
otherwise
There are two families as solution of the Einstein system. The Einstein constants for these two families are, respectively,
g) A new metric
Still assuming the same pattern, with , we find on
the invariant Einstein metric with
and
. The Einstein constant of this metric is
.
We define the class
.
A complete classification of the Einstein metrics for
is completely unknown. It is not even know if the number of such metrics is finite (the Bohn-Wang-Ziller conjecture).
In [11] and [10] we use the procedure described above in order to obtain new Einstein metrics on non-maximal A-type manifolds. Our notation will be
where
represents block-matrices of size
. All the entries in each block are equal, so that the metric is completely expressed by a reduced
matrix, which we denote by
.
Theorem 3.1. a) On The set of restrictions
produce two invariant non-Kähler Einstein. On the other hand the restrictions
do not produce any solution.
b) On i.e.
(
) we look for a
reduced matrix
with
(
),
otherwise
In this way we can produce two non-Kähler Einstein metrics.
c) On with
the invariant metric represented by the
matrix
is Einstein if, and only if, the same matrix represents an Einstein metric on
.
4. Results on the classification of Einstein metrics on
Gray and Hervella in [13] gave a complete classification of triples into sixteen classes for arbitrary almost Hermitian manifolds. San Martin-Negreiros discussed in [26] the case where
is a maximal flag manifold. They have proved that the invariant almost Hermitian structures on maximal flag manifolds can be divided only in three classes, namely
where the class
contains any invariant almost Hermitian structures.
In [26] it is proved that an invariant pair if and only if for all
-triple of roots
| (11) |
The next lemma characterizes the Hermitian structures belonging to (see [26]) for more details.
Lemma 4.1. A necessary and sufficient condition for an invariant pair to be in
is
-triple
.
In [11] or [10] the following result is proved:
Theorem 4.2. If for
, then this metric belongs to
.
This result leads us to conjecture that any invariant Einstein non-Kähler metric on is in
. One result supporting this conjecture is
Theorem 4.3. The space admits (up to scaling) precisely 3 classes of invariant Einstein metrics: The Kähler-Einstein [7], the 4 Arvanitoyeorgos's class [3], and the class of the normal metric [30].
[1] V. I. Arnold, Symplectization, Complexification and Mathematical Trinities, Fields Institute Communications, 24 (1999), 23-36. [ Links ]
[2] A. Arvanitoyeorgos, Invariant Einstein metrics on generalized flag manifolds, PhD thesis, Rochester University,1991. [ Links ]
[3] A. Arvanitoyeorgos, New invariant Einstein metrics on generalized flag manifolds, Trans. Amer. math.Soc., 337(1993), 981-995. [ Links ]
[4] A. Besse, Einstein Manifolds, Springer Verlag, 1987 [ Links ]
[5] C. Böhm, M. Wang and W. Ziller, A variational approach for compact homogeneous Einstein manifolds, Geometric and Functional Analysis 4 Vol.14, (2004) 681-733. [ Links ]
[6] M. Bordemann, M. Forger and H. Romer, Homogeneous Kähler manifolds:Paving the way towards supersymmetric sigma-models, Commun. Math. Physics,102, (1986), 605-647. [ Links ]
[7] A. Borel, Kählerian coset spaces of semi-simple Lie groups, Proc. Nat. Acad. Sci. USA 40 (1954), 1147-1151. [ Links ]
[8] É. Cartan, Sur les domaines bornés homogènes de l'espace des n variables complexes, Abhandl. Math. Sem. Hamburg, 11 (1935), 116-162 [ Links ]
[9] N. Cohen, C. J. C. Negreiros, M. Paredes, S. Pinzón, L. A. B. San Martin, f-structures on the classical flag manifold which admit (1,2)-symplectic metrics. Tôhoku Math. J. 57(2) (2005), 262-271. [ Links ]
[10] N. Cohen, E. C. F. dos Santos and C. J. C. Negreiros, Properties of Einstein metrics on flag manifolds, Preprint. State University of Campinas, 2005. [ Links ]
[11] Evandro C. F. dos Santos,Métricas de Einstein em varedades bandeira, PhD thesis. State University of Campinas, 2005. [ Links ]
[12] Evandro C. F. dos Santos, Caio J. C. Negreiros and Sofia Pinzòn, (1,2)-symplectic submanifols of a flag manifold, Preprint. State University of Campinas, 2005. [ Links ]
[13] A. Gray and L. M. Hervella, The sixteen classes of almost Hermitian manifolds and their linear invariants, Ann. Mat. Pura Appl. 123 (1980), 35-58. [ Links ]
[14] M.A.Guest, Geometry of maps between generalized flag manifolds, J. Differential Geometry, 25 (1987), 223-247 [ Links ]
[15] S. Helgason, Differential Geometry, Lie Groups and Symmetric Spaces, Academic Press, 1978. [ Links ]
[16] D.Hilbert, Die Grundlagen der Physik,Nachr. Akad. Wiss. Gött. 395-407 (1915). Reprinted in Math. Ann., 92 (1924),1-32. [ Links ]
[17] M. Kimura, Homogeneous Einstein Metrics on Certain Kähler C-spaces, Advanced Studies in Pure Mathematics. 18-I, 1990. Recent topics in Differential and analytic Geometry (1986), 303-320. [ Links ]
[18] S. Kobayashi and K. Nomizu, Foundations of differential geometry, Interscience Publishers, vol.2 (1969). [ Links ]
[19] Y. Matsushima, Remarks on Kähler-Einstein Manifolds, Nagoya Math. Journal 46, (1972), 161-173. [ Links ]
[20] J. Milnor, Curvatures of left invariant metrics on Lie groups, Adv. in Mathematics 21, (1976), 293-329. [ Links ]
[21] X. Mo and C. J. C. Negreiros, (1,2)-Symplectic structures on flag manifolds, Tohoku Math. J. 52 (2000), 271-282. [ Links ]
[22] Y. Mutö, On Einstein metrics, Journal of Differential geometry, 9 (1974), 521-530. [ Links ]
[23] M. Paredes, Aspectos da geometria complexa das variedades bandeira, Tese de Doutorado, Universidade Estadual de Campinas,2000. [ Links ]
[24] S. Pinzòn, Variedades bandeira,estruturas-f e mètricas (1,2)-simplèticas, Tese de Doutorado, Universidade Estadual de Campinas, 2003. [ Links ]
[25] Y. Sakane, Homogeneous Einstein metrics on flag manifolds, Lobatchevskii Journal of Mathematics, vol.4, (1999), 71-87. [ Links ]
[26] Luiz A. B. San Martin and Caio J. C. Negreiros, Invariant almost Hermitian structures on flag manifolds, Advances in Math., 178 (2003), 277-310. [ Links ]
[27] Luiz. A. B. San Martin and Rita. C. J. Silva, Invariant nearly-Kähler structures, preprint, State University of Campinas (2003). [ Links ]
[28] Rita de Cássia de J. Silva, Estruturas quase Hermitianas invariantes em espaços homogêneos de grupos semi-simples, Ph.D. thesis, State University of Campinas, 2003. [ Links ]
[29] M. Wang and W. Ziller, Existence and Non-Existence of Homogeneous Einstein Metrics, Invent. math., 84, (1986), 177-194. [ Links ]
[30] M. Wang and W. Ziller, On normal homogeneous Einstein metrics, Ann. Sci. Ecole norm. Sup.18 (1985), 563-633. [ Links ]
Evandro C. F. dos Santos
Department of Mathematics
Universidade Regional do Cariri
Av. Leão Sampaio s/n Km 4
Juazeiro do Norte-Ce
Cep 63040-000 - Brazil
evandrocfsantos@gmail.com
Caio J. C. Negreiros
Departament of Mathematics - IMECC - Unicamp
PO Box 6065 - Campinas - Brazil
caione@ime.unicamp.br
Recibido: 20 de octubre de 2005
Aceptado: 15 de noviembre de 2006