Symplectic Manifold - Special Cases and Generalizations

Special Cases and Generalizations

  • A symplectic manifold endowed with a metric that is compatible with the symplectic form is an almost Kähler manifold in the sense that the tangent bundle has an almost complex structure, but this need not be integrable. Symplectic manifolds are special cases of a Poisson manifold. The definition of a symplectic manifold requires that the symplectic form be non-degenerate everywhere, but if this condition is violated, the manifold may still be a Poisson manifold.
  • A multisymplectic manifold of degree k is a manifold equipped with a closed nondegenerate k-form. See F. Cantrijn, L. A. Ibort and M. de León, J. Austral. Math. Soc. Ser. A 66 (1999), no. 3, 303-330.
  • A polysymplectic manifold is a Legendre bundle provided with a polysymplectic tangent-valued -form; it is utilized in Hamiltonian field theory. See: G. Giachetta, L. Mangiarotti and G. Sardanashvily, Covariant Hamiltonian equations for field theory, Journal of Physics A32 (1999) 6629-6642; arXiv: hep-th/9904062.

Read more about this topic:  Symplectic Manifold

Famous quotes containing the words special and/or cases:

    “I think those Southern writers [William Faulkner, Carson McCullers] have analyzed very carefully the buildup in the South of a special consciousness brought about by the self- condemnation resulting from slavery, the humiliation following the War Between the States and the hope, sometimes expressed timidly, for redemption.”
    —Jimmy Carter (James Earl Carter, Jr.)

    “Lovers’ quarrels are not generally about money. Divorce cases generally are.”
    —Mason Cooley (b. 1927)