Decomposition in the large of two-forms of constant rank

Dibag, I.

doi:10.5802/aif.529

Cited by 4 publications

(5 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The following theorem states the necessary and sufficient conditions for the decomposition of a two-form of constant rank 2s in the large. ii) The representation of its normalization as a map w 1 : M → SO(2s)/U(s) arising from any trivialization of S w lifts to SO(2s) [3].…”

Section: The F Irst Obstruction: the Chern Class Of Qmentioning

confidence: 99%

See 1 more Smart Citation

Global Existence of Bi-Hamiltonian Structures on Orientable Three-Dimensional Manifolds

Efe¹,

Abadoglu²

2017

SIGMA

View full text Add to dashboard Cite

Abstract. In this work, we show that an autonomous dynamical system defined by a nonvanishing vector field on an orientable three-dimensional manifold is globally bi-Hamiltonian if and only if the first Chern class of the normal bundle of the given vector field vanishes. Furthermore, the bi-Hamiltonian structure is globally compatible if and only if the Bott class of the complex codimension one foliation defined by the given vector field vanishes.

show abstract

Section: The F Irst Obstruction: the Chern Class Of Qmentioning

confidence: 99%

“…ii) The representation of its normalization as a map w 1 : M → SO(2s)/U(s) arising from any trivialization of S w lifts to SO(2s) [3].…”

Section: The F Irst Obstruction: the Chern Class Of Qmentioning

confidence: 99%

Global Existence of Bi-Hamiltonian Structures on Orientable Three-Dimensional Manifolds

Efe¹,

Abadoglu²

2017

SIGMA

View full text Add to dashboard Cite

show abstract

“…represents A (n+2) d+1 , the space of normalized 2-forms in R n+2 of rank 2(d + 1), on which the Stiefel bundle V n+2,2(d+1) of orthonormal 2(d + 1)-frames in R n+2 induces a principal U (d + 1)-bundle (see [2,3]). …”

mentioning

confidence: 99%

“…For the existence and the decomposability of 2-forms see [2,3,8]. Finally, for 2-forms on spheres see [2,4].…”

mentioning

confidence: 99%

Pieri-type intersection formulas and primary obstructions for decomposing 2-forms

Sertöz

2001

Colloq. Math.

View full text Add to dashboard Cite

Abstract. We study the homological intersection behaviour for the Chern cells of the universal bundle of G(d, Q n ), the space of [d]-planes in the smooth quadric Q n in P n+1 over the field of complex numbers. For this purpose we define some auxiliary cells in terms of which the intersection properties of the Chern cells can be described. This is then applied to obtain some new necessary conditions for the global decomposability of a 2-form of constant rank. Introduction.In this article we study from a purely projective-geometric point of view the obstructions to globally decomposing a 2-form. It was shown by Dibag [3] that the vanishing of certain Chern classes is necessary for such a decomposition. We construct new classes whose nonvanishing implies the nonvanishing of the Chern classes. Moreover some vanishing patterns of these new classes imply the vanishing of the Chern class obstructions. This is achieved by studying the intersection structure of the integral homology generated by the Chern cells. Our methods are purely geometric and determine the required products up to a nonzero multiplicative constant. However this suffices for our purposes since we eventually check for vanishing of obstructions. In the case of maximal planes these coefficients can be explicitly calculated. This is done by Hiller and Boe [7] who consider the case of type B maximal isotropic Grassmannians. Type D (which is a consequence of the result in type B) appeared in [9]. The results of [7] are further reproved by Pragacz and Ratajski [11] by using divided differences. Recently similar calculations in type B were done by Sottile [14]. In the nonmaximal case these calculations are due to Pragacz and Ratajski (see [12]). However to adopt these general formulas for our cases would lead to complicated combinatorial formulas. By checking only nonvanishing conditions we are able to present a purely geometrical argument which suffices for our results.We denote by G(d, Q n ) the space of complex projective [d]-planes lying in the smooth quadric hypersurface Q n of P n+1 . Dibag has shown that G(d, Q n )

show abstract

“…Analogously, if w is a 2-form on M of constant rank 2s, the union of the 2i,-planes on which w is of maximal rank forms a sub-bundle Sw of T(M), and the orthogonal-component of w (regarded as a nonsingular, skew-adjoint transformation on Sw) defines a normalized almost-complex ^-substructure (e.g. refer to [3]). We thus have: Existence of an almost-complex s-substructure <=> Existence of a normalized almost-complex ^--substructure <=> Existence of a 2-form on M of constant rank 2s.…”

mentioning

confidence: 99%