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
RESUMENMo and Negreiros [14], by using moving frames and tournaments, showed explicitly the existence of an -dimensional family of invariant (1,2)-symplectic metrics on . This family corresponds to the family of the parabolic almost complex structures on . 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 , different to the parabolic one.
Eells and Sampson [8] proved that if is a holomorphic map between Kähler manifolds then is harmonic. This result was generalized by Lichnerowicz (see [12] or [20]) as follows: Let and be almost Hermitian manifolds with cosymplectic and (1,2)-symplectic, then any -holomorphic map is harmonic.
If we want to obtain harmonic maps, , from a closed Riemann surface to a full flag manifold by the Lichnerowicz theorem, we must study (1,2)-symplectic metrics on 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 it is natural to begin by studing its invariant almost complex structures. Borel and Hirzebruch [5] proved that there are -invariant almost complex structures on . This number is the same number of tournaments with 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 and tournaments with 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 on is integrable if and only if the tournament associated to is isomorphic to the canonical tournament (the canonical tournament with players, , is defined by if and only if ).
Borel proved the existence of an -dimensional family of invariant Kähler metrics on for each invariant complex structure on (see [2] or [4]). Eells and Salamon [9] proved that any parabolic structure on admits a (1,2)-symplectic metric. Mo and Negreiros [14] showed explicitly that there is an -dimensional family of invariant (1,2)-symplectic metrics for each parabolic structure on .
In this paper, we characterize new -parametric families of (1,2)-symplectic invariant metrics on , different to the Kähler and parabolic ones. More precisely, we obtain explicitly different -dimensional families of (1,2)-symplectic invariant metrics, for each . Each of them corresponds to a different class of non-integrable invariant almost complex structure on . These metrics are used to produce new examples of harmonic maps , using the previous result by Lichnerowicz.
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