Symplectic vector space

About: Symplectic vector space is a research topic. Over the lifetime, 2048 publications have been published within this topic receiving 53456 citations.

Symplectic homogeneous spaces

Abstract: It is proved in this paper that for a given simply connected Lie group G with Lie algebra g, every left-invariant closed 2-form induces a symplectic homogeneous space. This fact generalizes the results in [7] and [12] that if H1l(g)= H2(g) =0, then the most general symplectic homogeneous space covers an orbit in the dual of the Lie algebra S. A one-to-one correspondence can be established between the orbit space of equivalent classes of 2-cocycles of a given Lie algebra and the set of equivalent classes of simply connected symplectic homogeneous spaces of the Lie group. Lie groups with left-invariant symplectic structure cannot be semisimple; hence such groups of dimension four have to be solvable, and connected unimodular groups with left-invariant symplectic structure are solvable [4]. 1. Symplectic manifolds. Let M be a 2n-dimensional connected differentiable manifold. A symplectic structure on M is defined by a closed differential 2form c) which is everywhere of maximal rank. Such a form is called a symplectic form of the symplectic structure defined on M. On a symplectic manifold M, a one-to-one map from the space of vector fields X(M) onto the space of linear differential forms D1(M) can be defined as follows. If X is a vector field, the map x "i(X)w (where i(X)w denotes the interior product of X with wo) is a bijective map from X(M) onto DI(M). In fact, at each point x c M, this map from TX(M) onto 7* (M) is given by the nonsingular bilinear form o) A classical theorem attributed to Darboux [11] states that for an n-dimensional manifold M with a closed 2-form w) of rank exactly p everywhere there can be introduced about every point a system of coordinates x1,*. *, xn-P, y * yP, in terms of which the local representation of cw becomes Received by the editors April 23, 1973 and, in revised form, July 10, 1973. AMS (MOS) subject classifications (1970). Primary 53C30; Secondary 57F15, 22E25.

Contact Geometry and Nonlinear Differential Equations

Abstract: Introduction Part I. Symmetries and Integrals: 1. Distributions 2. Ordinary differential equations 3. Model differential equations and Lie superposition principle Part II. Symplectic Algebra: 4. Linear algebra of symplectic vector spaces 5. Exterior algebra on symplectic vector spaces 6. A Symplectic classification of exterior 2-forms in dimension 4 7. Symplectic classification of exterior 2-forms 8. Classification of exterior 3-forms on a 6-dimensional symplectic space Part III. Monge-Ampere Equations: 9. Symplectic manifolds 10. Contact manifolds 11. Monge-Ampere equations 12. Symmetries and contact transformations of Monge-Ampere equations 13. Conservation laws 14. Monge-Ampere equations on 2-dimensional manifolds and geometric structures 15. Systems of first order partial differential equations on 2-dimensional manifolds Part IV. Applications: 16. Non-linear acoustics 17. Non-linear thermal conductivity 18. Meteorology applications Part V. Classification of Monge-Ampere Equations: 19. Classification of symplectic MAEs on 2-dimensional manifolds 20. Classification of symplectic MAEs on 2-dimensional manifolds 21. Contact classification of MAEs on 2-dimensional manifolds 22. Symplectic classification of MAEs on 3-dimensional manifolds.

Symplectic structure on ruled surfaces and a generalized adjunction formula

Abstract: In this section, we state the theorems needed in our paper. Recently, Seiberg and Witten ([SW], [Wi]) have introduced a new set of 4-manifold invariants. These invariants are in similar spirit to Donaldson invariants but much easier to handle. Various longstanding conjectures including the Thom conjecture are proved using Seiberg-Witten invariants. An important ingredient in the proof of the Thom conjecture by Kronheimer and Mrowka is the wall crossing formula for manifolds with b1 = 0. Seiberg-Witten invariants take on a very simple form for Kahler surfaces ([Wi], [B], [FM1]). All the basic classes are explicitly known and in particular, the anticanonical bundle is always a basic class. A large part of this story is generalized to symplectic manifolds by Taubes who ([T1], [T2], [T3], [T4]) proved several remarkable theorems on Seiberg-Witten invariants of symplectic four-manifolds. Recall that every symplectic manifold has a complex line bundle, K (called the canonical bundle), which is canonical up to isomorphism. The first theorem of Taubes is

Hermitian symplectic geometry and extension theory

Abstract: Here we give brief account of Hermitian symplectic spaces, showing that they are intimately connected to symmetric as well as self-adjoint extensions of a symmetric operator. Furthermore, we find an explicit parametrization of the Lagrange Grassmannian in terms of the unitary matrices . This allows us to explicitly describe all self-adjoint boundary conditions for the Schrodinger operator on the graph in terms of a unitary matrix. We show that the asymptotics of the scattering matrix can be expressed simply in terms of this unitary matrix.