On symplectic periods of the discrete spectrum ofGL 2n

We provide a formula for the symplectic period of an automorphic form in the discrete spectrum ofGL _2n.It is a generalization of a formula of Jacquet and Rallis.     

On extensions of a symplectic class

Let F be a fibration on a simply-connected base with symplectic fiber (M,ω). Assume that the fiber is nilpotent and T2k-separable for some integer k or a nilmanifold. Then our main theorem, Theorem 1.8, gives a necessary and sufficient condition for the cohomology class [ω] to extend to a cohomology class of the total space of F. This allows us to describe Thurston?s criterion for a symplectic fibration to admit a compatible symplectic form in terms of the classifying map for the underlying fibration. The obstruction due to Lalond and McDuff for a symplectic bundle to be Hamiltonian is also rephrased in the same vein. Furthermore, with the aid of the main theorem, we discuss a global nature of the set of the homotopy equivalence classes of fibrations with symplectic fiber in which the class [ω] is extendable.     

On the asphericity of a symplectic

C. H. Taubes asked whether a closed (i.e. compact and without boundary) connected oriented three-dimensional manifold whose product with a circle admits a symplectic structure must fiber over a circle. An affirmative answer to Taubes' question would imply that any such manifold either is diffeomorphic to the product of a two-sphere with a circle or is irreducible and aspherical. In this paper, we prove that this implication holds up to connect sum with a manifold which admits no proper covering spaces with finite index. It is pointed out that Thurston's geometrization conjecture and known results in the theory of three-dimensional manifolds imply that such a manifold is a three-dimensional sphere. Hence, modulo the present conjectural picture of three-dimensional manifolds, we have shown that the stated consequence of an affirmative answer to Taubes' question holds.

     

On the Iwasawa decomposition of a symplectic matrix

We consider the computation of the Iwasawa decomposition of a symplectic matrix via the QR factorization. The algorithms presented improve on the method recently described by T.-Y. Tam in [Computing Iwasawa decomposition of a symplectic matrix by Cholesky factorization, Appl. Math. Lett. (in press) doi:10.1016/j.aml.2006.03.001].     

On symplectic cobordisms

In this note we make several observations concerning symplectic cobordisms. Among other things we show that every contact 3-manifold has infinitely many concave symplectic fillings and that all overtwisted contact 3-manifolds are “symplectic cobordism equivalent”. Received: 26 March 2001 / Revised version: 1 May 2001 / Published online: 28 February 2002     

On symplectic transformations

This is an English translation¹ of the Ph.D. thesis ‘Over symplectische transformaties’ that Tonny Albert Springer, ‘born in’s-Gravenhage in 1926’, submitted as thesis for – as is stated on the original frontispiece – the degree of doctor in mathematics and physics at Leiden University on the authority of the rector magnificus Dr. J.H. Boeke, professor in the faculty of law, to be defended against the objections of the Faculty of Mathematics and Physics on Wednesday October 17 1951 at 4 p.m., with promotor Prof. dr. H. D. Kloosterman.     

On adjacencies of orbits of a symplectic group action on the manifold of complete flags in a symplectic space

Translated from Itogi Nauki i Tekhniki, Seriya Sovremenaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 20, Topologia-3, 1994.     

On the trivectors of a 6-dimensional symplectic vector space

Let V be a 6-dimensional vector space over a field F, let f be a nondegenerate alternating bilinear form on V and let Sp(V,f)≅Sp₆(F) denote the symplectic group associated with (V,f). The group GL(V) has a natural action on the third exterior power ?³V of V and this action defines five families of nonzero trivectors of V (four of whose are orbits for any choice of F). In this paper, we divide three of these five families into orbits for the action of Sp(V,f)⊆GL(V) on ?³V.     

On the complex structure of symplectic quotients

Science China Mathematics - Let K be a compact group. For a symplectic quotient Mλ of a compact Hamiltonian Kähler K-manifold, we show that the induced complex structure on Mλ is...     

On symplectic half-flat manifolds

We construct examples of symplectic half-flat manifolds on compact quotients of solvable Lie groups. We prove that the Calabi-Yau structures are not rigid in the class of symplectic half-flat structures. Moreover, we provide an example of a compact 6-dimensional symplectic half-flat manifold whose real part of the complex volume form is d-exact. Finally we discuss the 4-dimensional case. This work was supported by the Projects M.I.U.R. “Geometric Properties of Real and Complex Manifolds”, “Riemannian Metrics and Differentiable Manifolds” and by G.N.S.A.G.A. of I.N.d.A.M.     

On quadratic symplectic mappings

Dedicated to M. Hénon on the occasion of his 60th birthday     

On the geography of symplectic 6-manifolds

The article investigates the geography of closed, connected and simply connected, six-dimensional manifolds. It is proved that any triple of integers satisfying some necessary arithmetical restrictions occurs as the Chern triple of such a manifold. The main tools used for producing the examples are the symplectic connected sum and the symplectic blow-up. Received: 28 May 1998 / Revised version: 22 January 1999     

On the symplectic phase space of KdV

We prove that the Birkhoff map for KdV constructed on can be interpolated between and . In particular, the symplectic phase space can be described in terms of Birkhoff coordinates.