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:
“Fashions change, and with the new psychoanalytical perspective of the postwar period [WWII], child rearing became enshrined as the special responsibility of mothers ... any shortcoming in adult life was now seen as rooted in the failure of mothering during childhood.”
—Sylvia Ann Hewitt (20th century)
“We noticed several other sandy tracts in our voyage; and the course of the Merrimack can be traced from the nearest mountain by its yellow sand-banks, though the river itself is for the most part invisible. Lawsuits, as we hear, have in some cases grown out of these causes. Railroads have been made through certain irritable districts, breaking their sod, and so have set the sand to blowing, till it has converted fertile farms into deserts, and the company has had to pay the damages.”
—Henry David Thoreau (1817–1862)