## 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)

### MSC:

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

### Keywords:

contact manifold; shape of a manifold; symplectic manifold
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### References:

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