In mathematics, a Delzant polytope is a
convex polytope
A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the wo ...
in
such that for each vertex
, exactly
edges meet at
(that is, it is a
simple polytope
In geometry, a -dimensional simple polytope is a -dimensional polytope each of whose vertices are adjacent to exactly edges (also facets). The vertex figure of a simple -polytope is a -simplex.
Simple polytopes are topologically dual to s ...
), and there are integer vectors parallel to these edges forming a
-basis of
.
Delzant's theorem, introduced by , classifies effective Hamiltonian torus actions on compact connected
symplectic manifold
In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M , equipped with a closed nondegenerate differential 2-form \omega , called the symplectic form. The study of symplectic manifolds is called sy ...
s by the image of the associated
moment map, which is a Delzant polytope.
The theorem states that there is a bijective correspondence between symplectic toric manifolds (up to torus-equivariant symplectomorphism) and ''Delzant polytopes''. More precisely, the moment polytope of every symplectic toric manifold is a Delzant polytope, every Delzant polytope is the moment polytope of such a manifold, and any two such manifolds with equivalent moment polytopes (up to translations and
transformations) admit a torus-equivariant symplectomorphism between them.
References
*
Symplectic geometry
Theorems in differential geometry
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