Families of (1,2)-Symplectic M etrics on Full Flag Manifolds
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∗ †
1991 Mathematical Subject Classification. Primary 53C55; Secondary 58E20, 05C20. Supported by CAPES–Brazil and COLCIENCIAS–Colombia
பைடு நூலகம்
1
If we like to obtain harmonic maps, φ : M 2 → F (n), from a closed Riemannian surface M 2 to a full flag manifold F (n), by the Lichnerowicz theorem we have to study (1,2)–symplectic metrics on F (n) because a Riemann surface is a K¨ ahler manifold and we know that a K¨ ahler manifold is a cosymplectic manifold (see [19] or [10]). To study the invariant Hermitian geometry of F (n) it is natural to begin by studing its invariant almost complex structures. Borel and Hirzebruch n [4], proved that there are 2( 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 [12] or [5]). There is a natural identification between almost complex structures on F (n) and tournaments with n players (see [14] or [5]). 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 [5], proved that a almost complex structure J on F (n) is integrable if and only if the associated tournament to J is isomorphic to the canonical tournament (the canonical tournament with n players, {1, 2, . . . , n}, is defined by i → j if and only if i < j ). Borel proved that exits a (n − 1)–dimensional family of invariant K¨ ahler metrics on F (n) for each invariant complex structure on F (n) (see [1] or [3]). Eells and Salamon [8], proved that any parabolic structure on F (n) admits a (1,2)–symplectic metric. Mo and Negreiros [13], showed explicitly that there is a n–dimensional family of invariant (1,2)–symplectic metrics for each parabolic structure on F (n). In this paper, we characterize new n–parametrical families of (1,2)–symplectic invariant metrics on F (n), different to the K¨ ahler and parabolic. More precisely, we obtain explicitly n − 3 different n–dimensional families of (1,2)– symplectic invariant metrics, for each n ≥ 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 φ : M 2 → F (n), using the result of Lichnerowicz above. This paper is part of the author’s Doctoral Thesis [17]. I wish to thank 2
n−times
where T = U (1) × · · · × U (1) is a maximal torus in U (n).
n−times
Let p be the tangent space of F (n) in (T ). An invariant almost complex structure on F (n) is a linear map J : p → p such that J 2 = −I . A tournament or n–tournament T , consists of a finite set T = {p1 , p2 , . . . , pn } of n players, together with a dominance relation, →, that assigns to every pair of players a winner, i.e. pi → pj or pj → pi . If pi → pj then we say that pi beats pj . A tournament T may be represented by a directed graph in which T is the set of vertices and any two vertices are joined by an oriented edge. Let T1 be a tournament with n players {1, . . . , n} and T2 another tournament with m players {1, . . . , m}. A homomorphism between T1 and T2 is a mapping φ : {1, . . . , n} → {1, . . . , m} such that (2.3)
arXiv:math/0010219v1 [math.DG] 23 Oct 2000
Families of (1,2)–Symplectic Metrics on Full Flag Manifolds∗
Marlio Paredes†
Abstract We obtain new families of (1,2)–symplectic invariant metrics on the full complex flag manifolds F (n). For n ≥ 5, we characterize n − 3 different n–dimensional families of (1,2)–symplectic invariant metrics on F (n). Any of these families corresponds to a different class of non–integrable invariant almost complex structure on F (n).
2
Preliminaries
A full flag manifold is defined by (2.1) F (n) = {(L1 , . . . , Ln ) : Li is a subspace of Cn , dimC Li = 1, Li ⊥Lj }. The unitary group U (n) acts transitively on F (n). Using this action we obtain an algebraic description for F (n): (2.2) F (n) = U (n) U (n) = , T U (1) × · · · × U (1)
1
Introduction
Recently Mo and Negreiros [13], by using moving frames and tournaments, showed explicitly the existence of a n–dimensional family of invariant (1,2)– U (n) symplectic metrics on F (n) = U (1)×···× . This family corresponds to the U (1) 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 structure on F (n) different to the parabolic. Eells and Sampson [7], proved that if φ : M → N is a holomorphic map between K¨ ahler manifolds then φ is harmonic. This result was generalized by Lichnerowicz (see [11] or [19]) as follows: Let (M, g, J1) and (N, h, J2 ) be almost Hermitian manifolds with M cosymplectic and N (1,2)–symplectic. Then any holomorphic map φ : (M, g, J1 ) → (N, h, J2 ) is harmonic.
my advisor Professor Caio Negreiros for his right advise. I would like to thank Professor Xiaohuan Mo for his helpful comments and dicussions on this work.
1991 Mathematical Subject Classification. Primary 53C55; Secondary 58E20, 05C20. Supported by CAPES–Brazil and COLCIENCIAS–Colombia
பைடு நூலகம்
1
If we like to obtain harmonic maps, φ : M 2 → F (n), from a closed Riemannian surface M 2 to a full flag manifold F (n), by the Lichnerowicz theorem we have to study (1,2)–symplectic metrics on F (n) because a Riemann surface is a K¨ ahler manifold and we know that a K¨ ahler manifold is a cosymplectic manifold (see [19] or [10]). To study the invariant Hermitian geometry of F (n) it is natural to begin by studing its invariant almost complex structures. Borel and Hirzebruch n [4], proved that there are 2( 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 [12] or [5]). There is a natural identification between almost complex structures on F (n) and tournaments with n players (see [14] or [5]). 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 [5], proved that a almost complex structure J on F (n) is integrable if and only if the associated tournament to J is isomorphic to the canonical tournament (the canonical tournament with n players, {1, 2, . . . , n}, is defined by i → j if and only if i < j ). Borel proved that exits a (n − 1)–dimensional family of invariant K¨ ahler metrics on F (n) for each invariant complex structure on F (n) (see [1] or [3]). Eells and Salamon [8], proved that any parabolic structure on F (n) admits a (1,2)–symplectic metric. Mo and Negreiros [13], showed explicitly that there is a n–dimensional family of invariant (1,2)–symplectic metrics for each parabolic structure on F (n). In this paper, we characterize new n–parametrical families of (1,2)–symplectic invariant metrics on F (n), different to the K¨ ahler and parabolic. More precisely, we obtain explicitly n − 3 different n–dimensional families of (1,2)– symplectic invariant metrics, for each n ≥ 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 φ : M 2 → F (n), using the result of Lichnerowicz above. This paper is part of the author’s Doctoral Thesis [17]. I wish to thank 2
n−times
where T = U (1) × · · · × U (1) is a maximal torus in U (n).
n−times
Let p be the tangent space of F (n) in (T ). An invariant almost complex structure on F (n) is a linear map J : p → p such that J 2 = −I . A tournament or n–tournament T , consists of a finite set T = {p1 , p2 , . . . , pn } of n players, together with a dominance relation, →, that assigns to every pair of players a winner, i.e. pi → pj or pj → pi . If pi → pj then we say that pi beats pj . A tournament T may be represented by a directed graph in which T is the set of vertices and any two vertices are joined by an oriented edge. Let T1 be a tournament with n players {1, . . . , n} and T2 another tournament with m players {1, . . . , m}. A homomorphism between T1 and T2 is a mapping φ : {1, . . . , n} → {1, . . . , m} such that (2.3)
arXiv:math/0010219v1 [math.DG] 23 Oct 2000
Families of (1,2)–Symplectic Metrics on Full Flag Manifolds∗
Marlio Paredes†
Abstract We obtain new families of (1,2)–symplectic invariant metrics on the full complex flag manifolds F (n). For n ≥ 5, we characterize n − 3 different n–dimensional families of (1,2)–symplectic invariant metrics on F (n). Any of these families corresponds to a different class of non–integrable invariant almost complex structure on F (n).
2
Preliminaries
A full flag manifold is defined by (2.1) F (n) = {(L1 , . . . , Ln ) : Li is a subspace of Cn , dimC Li = 1, Li ⊥Lj }. The unitary group U (n) acts transitively on F (n). Using this action we obtain an algebraic description for F (n): (2.2) F (n) = U (n) U (n) = , T U (1) × · · · × U (1)
1
Introduction
Recently Mo and Negreiros [13], by using moving frames and tournaments, showed explicitly the existence of a n–dimensional family of invariant (1,2)– U (n) symplectic metrics on F (n) = U (1)×···× . This family corresponds to the U (1) 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 structure on F (n) different to the parabolic. Eells and Sampson [7], proved that if φ : M → N is a holomorphic map between K¨ ahler manifolds then φ is harmonic. This result was generalized by Lichnerowicz (see [11] or [19]) as follows: Let (M, g, J1) and (N, h, J2 ) be almost Hermitian manifolds with M cosymplectic and N (1,2)–symplectic. Then any holomorphic map φ : (M, g, J1 ) → (N, h, J2 ) is harmonic.
my advisor Professor Caio Negreiros for his right advise. I would like to thank Professor Xiaohuan Mo for his helpful comments and dicussions on this work.