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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.
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TL;DR: In this paper, the concept of symplectic arc length for curves is introduced and the authors construct an adapted symplectic Frenet frame and define 2n - 1 local differential invariants that they call symplectic curvatures of the curve.
Abstract: We consider curves in ℝ2n endowed with the standard symplectic structure. We introduce the concept of symplectic arc length for curves. We construct an adapted symplectic Frenet frame and we define 2n - 1 local differential invariants that we call symplectic curvatures of the curve. We prove that up to a rigid symplectic motion of ℝ2n, there exists a unique curve with prescribed symplectic curvatures. We characterize the curves in ℝ4 with constant symplectic curvatures.
14 citations
TL;DR: In this paper, a class of affine symmetric spaces with a non-Abelian solvable transvection group is defined, where the underlying manifold M of each element (M, ▿) belonging to this class is diffeomorphic to Rn.
Abstract: A symplectic is a symmetric space endowed with a symplectic structure which is invariant by the symmetries. We give here a classification of four-dimensional symplectic which are simply connected. This classification reveals a remarkable class of affine symmetric spaces with a non-Abelian solvable transvection group. The underlying manifold M of each element (M, ▿) belonging to this class is diffeomorphic to Rnwith the property that every tensor field on M invariant by the transvection group is constant; in particular, ▿ is not a metric connection. This classification also provides examples of nonflat affine symmetric connections on Rnwhich are invariant under the translations. By considering quotient spaces, one finds examples of locally affine symmetric tori which are not globally symmetric.
14 citations
TL;DR: Bouchet as discussed by the authors showed that every Eulerian multimatroid is representable with a symplectic vector space over GF(2) and adapted the construction to symplectic matroids.
14 citations
TL;DR: In this article, the invariant moment map is shown to be equidimensional and the categorical quotient is a fibration over an affine space with rational generic fibers.
14 citations
TL;DR: In this paper, the authors investigated connections between the number of focal points of their recessive solutions of the Riccati matrix difference equations and the comparative index theory for discrete symplectic systems.
Abstract: We present new relations connecting the number of focal points of conjoined bases of eventually disconjugate symplectic difference systems with different coefficient matrices which obey the so-called majorant condition at For the case of controllable (near ) symplectic systems we investigate connections between the number of focal points of their recessive solutions. The results of the paper are based on the concept of minimal and maximal solutions of the associated Riccati matrix difference equations and the comparative index theory for discrete symplectic systems.
14 citations