New invariants of open symplectic and contact manifolds. (English) Zbl 0733.58011

Let (X,\(\omega\)) be a symplectic manifold of dimension 2n, f: \(M\to X\) a Lagrangian embedding (i.e., \(f^*\omega =0)\) of a closed manifold M of dimension n. Let \(\omega =d\lambda\) and \(\alpha: H^ 1(X,{\mathbb{R}})\to H^ 1(M,{\mathbb{R}})\) be a fixed homomorphism. The subset \(I(X/M,\alpha)\subset H^ 1(M,{\mathbb{R}})\) consisting of all \(z=[f^*\lambda]\in H^ 1(M,{\mathbb{R}})\) where \(f^*=\alpha\) (and f is an appropriate variable Lagrange embedding) is called an (M,\(\alpha\))-shape of X. The shapes can serve for obstructions to the existence of a symplectical embedding in a given homotopy class.
For a contact (2n-1)-dimensional manifold U defined by a 1-form \(\mu\) (where \(\mu \wedge (d\mu)^{n-1}\neq 0)\), an analogous concept of contact (M,\(\alpha\))-shape can be defined by the use of the contact form \(\omega =d(t\mu)\) on the product manifold \(U\times {\mathbb{R}}_+\). (A self- contained version of this concept is introduced, too.) The paper is concluded by classification of contactomorphisms of 3-dimensional solid tori.
Reviewer: J.Chrastina (Brno)


37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
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