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Revista de la Unión Matemática Argentina
versión impresa ISSN 0041-6932versión On-line ISSN 1669-9637
Rev. Unión Mat. Argent. v.47 n.2 Bahía Blanca jul./dic. 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].
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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