BOLETÍN ELECTRÓNICO CIENTÍFICO
DEL NODO BRASILERO
DE INVESTIGADORES COLOMBIANOS
Número 3(Artículo 7), 2001

TÍTULO
FAMILIES OF (1,2)-SYMPLECTIC METRICS ON FULL FLAG MANIFOLDS

TIPO: Artículo aceptado para publicación en International Journal of Mathematics and Mathematical Sciences

AUTOR: Marlio Paredes Gutierrez mparedes@uis.edu.co

IDIOMA: Inglés

DIRECCIÓN PARA CONTACTO
Escuela de Matemáticas
Universidad Industrial de Santander
A.A. 678, Bucaramanga, Colombia

ENTIDADES QUE FINANCIARON LA INVESTIGACIÓN: CAPES, COLCIENCIAS

PALABRAS CLAVE: (1,2)-symplectic metrics, flag manifolds, tournaments, harmonic maps. Clasificación de la AMS: Primary 53C15, 53C55; Secondary 14M15, 05C20, 58E20

RESUMEN

Mo and Negreiros [14], by using moving frames and tournaments, showed explicitly the existence of an $ n$-dimensional family of invariant (1,2)-symplectic metrics on $ F(n) = U(n)/(U(1) \times \cdots \times U(1))$. This family corresponds to the family of the parabolic almost complex structures on $ F(n)$. In this paper we study the existence of other families of invariant (1,2)-symplectic metrics corresponding to classes of non-integrable invariant almost complex structures on $ F(n)$, different to the parabolic one.

Eells and Sampson [8] proved that if $ \phi \colon M \rightarrow N$ is a holomorphic map between Kähler manifolds then $ \phi$ is harmonic. This result was generalized by Lichnerowicz (see [12] or [20]) as follows: Let $ (M,g,J_1)$ and $ (N,h,J_2)$ be almost Hermitian manifolds with $ M$ cosymplectic and $ N$ (1,2)-symplectic, then any $ \pm$-holomorphic map $ \phi \colon (M,g,J_1) \rightarrow (N,h,J_2)$ is harmonic.

If we want to obtain harmonic maps, $ \phi \colon M^2 \rightarrow F(n)$, from a closed Riemann surface $ M^2$ to a full flag manifold $ F(n)$ by the Lichnerowicz theorem, we must study (1,2)-symplectic metrics on $ F(n)$ because a Riemann surface is a Kähler manifold and we know that a Kähler manifold is a cosymplectic manifold (see [20] or [11]).

To study the invariant Hermitian geometry of $ F(n)$ it is natural to begin by studing its invariant almost complex structures. Borel and Hirzebruch [5] proved that there are $ 2^{\binom{n}{2}}$ $ U(n)$-invariant almost complex structures on $ F(n)$. This number is the same number of tournaments with $ n$ players or nodes. A tournament is a digraph in which any two nodes are joined by exactly one oriented edge (see [13] or [6]). There is a natural identification between almost complex structures on $ F(n)$ and tournaments with $ n$ players (see [15] or [6]).

Tournaments can be classified in isomorphism classes. In this classification, one of these classes corresponds to the integrable structures and the other ones correspond to non-integrable structures. Burstall and Salamon [6] proved that an almost complex structure $ J$ on $ F(n)$ is integrable if and only if the tournament associated to $ J$ is isomorphic to the canonical tournament (the canonical tournament with $ n$ players, $ \{1,2,\ldots,n\}$, is defined by $ i \rightarrow j$ if and only if $ i < j$).

Borel proved the existence of an $ (n-1)$-dimensional family of invariant Kähler metrics on $ F(n)$ for each invariant complex structure on $ F(n)$ (see [2] or [4]). Eells and Salamon [9] proved that any parabolic structure on $ F(n)$ admits a (1,2)-symplectic metric. Mo and Negreiros [14] showed explicitly that there is an $ n$-dimensional family of invariant (1,2)-symplectic metrics for each parabolic structure on $ F(n)$.

In this paper, we characterize new $ n$-parametric families of (1,2)-symplectic invariant metrics on $ F(n)$, different to the Kähler and parabolic ones. More precisely, we obtain explicitly $ n-3$ different $ n$-dimensional families of (1,2)-symplectic invariant metrics, for each $ n \geq 5$. Each of them corresponds to a different class of non-integrable invariant almost complex structure on $ F(n)$. These metrics are used to produce new examples of harmonic maps $ \phi \colon M^2 \rightarrow F(n)$, using the previous result by Lichnerowicz.

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