Symplectomorphism

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In mathematics, a symplectomorphism is an isomorphism in the category of symplectic manifolds.

Contents 1 Formal definition 2 Flows 3 Comparison with Riemannian geometry 4 The group of (Hamiltonian) symplectomorphisms 5 Quantizations 6 Arnold conjecture 7 References

[edit] Formal definition

Specifically, let (M₁, ω₁) and (M₂, ω₂) be symplectic manifolds. A map

f : M₁ → M₂

is a symplectomorphism if it is a diffeomorphism and the pullback of ω₂ under f is equal to ω₁:

Examples of symplectomorphisms include the canonical transformations of classical mechanics and theoretical physics, the flow associated to any Hamiltonian function, and the map on cotangent bundles induced by any diffeomorphism of manifolds.

[edit] Flows

Any smooth function on a manifold gives rise to a Hamiltonian vector field, which are special cases of symplectic vector fields. Flows of the latter give rise to symplectomorphisms. Since symplectomorphisms preserve volume, Liouville's theorem in Hamiltonian mechanics follows. Symplectomorphisms that arise from Hamiltonian vector fields are known as Hamiltonian symplectomorphisms.

Since

{H,H} = X_H(H) = 0

the flow of a Hamiltonian vector field also preserves H. In physics this is interpreted as the law of conservation of energy.

If the first Betti number of a connected symplectic manifold is zero, symplectic and Hamiltonian vector fields coincide, so the notions of Hamiltonian isotopy and symplectic isotopy of symplectomorphisms coincide.

The equations for a geodesic may be formulated as a Hamiltonian flow.

[edit] Comparison with Riemannian geometry

Unlike general Riemannian manifolds, symplectic manifolds are not very rigid: All symplectic manifolds are locally isomorphic by Darboux's theorem. In contrast, isometries in Riemannian geometry must preserve curvature tensor, which is a local invariant. Every function gives rise to a Hamiltonian diffeomorphism, so their symplectomorphism groups are always infinite dimensional. Riemannian manifolds generally have no nontrivial symmetries, and those with large symmetry groups are special.

[edit] The group of (Hamiltonian) symplectomorphisms

The group of symplectomorphisms from a manifold back onto itself forms an infinite-dimensional Lie group, of which the group of Hamiltonian symplectomorphisms forms a subgroup. The corresponding Lie algebra consists of symplectic vector fields (of which the Hamiltonian vector fields form a subalgebra). Poisson bracket on smooth functions on the manifold is another closely related lie algebra.

Groups of Hamiltonian diffeomorphisms are simple by a theorem of Banyaga. They have a natural geometry given by the Hofer norm. Gromov's theory of pseudoholomorphic curves can be used to compute the homotopy type of the symplectomorphism group for certain simple symplectic four-manifolds such as the product of spheres.

[edit] Quantizations

Representations of finite-dimensional subgroups of the group of symplectomorphisms (after -deformations, in general) on Hilbert spaces are called quantizations. When the Lie group is the one defined by a Hamiltonian, it is called a "quantization by energy". The corresponding operator from the Lie algebra to the Lie algebra of continuous linear operators is also sometimes called the quantization; this is a more common way of looking at it in physics. See Weyl quantization, geometric quantization, non-commutative geometry.

[edit] Arnold conjecture

A celebrated conjecture of V. I. Arnold relates the minimum number of fixed points for a Hamiltonian symplectomorphism f on M, in case M is a closed manifold, to Morse theory. More precisely, the conjecture states that f has at least as many fixed points as the number of critical points a smooth function on M must have (understood as for a generic case, Morse functions, for which this is a definite finite number which is at least 2).

It is known that this would follow from the Arnold-Givental conjecture, which is a statement on Lagrangian submanifolds. It is proven in many cases by the construction of symplectic Floer homology.

[edit] References

• Dusa McDuff and D. Salamon: Introduction to Symplectic Topology (1998) Oxford Mathematical Monographs, ISBN 0-19-850451-9.
• Ralph Abraham and Jarrold E. Marsden, Foundations of Mechanics, (1978) Benjamin-Cummings, London ISBN 0-8053-0102-X See section 3.2.

Symplectomorphism groups:

• Gromov, M. Pseudoholomorphic curves in symplectic manifolds. Invent. Math. 82 (1985), no. 2, 307--347.
• Polterovich, Leonid. The geometry of the group of symplectic diffeomorphism. Basel ; Boston : Birkhauser Verlag, 2001.