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Coisotropic submanifolds and dual pairs. (English) Zbl 1326.53119
Consider a finite-dimensional manifold $$M$$. Denote by $$I$$ the unit interval $$[0, 1]$$. The space of differentiable paths on $$M$$ is denoted by $$P M := C^1 (I, M)$$, and the space of bundle maps $$T I \to T^*M$$ with continuously differentiable base map and continuous fiber map is denoted by $$T^*P M$$. The space $$P M$$ is considered as a Banach manifold and $$T^*P M$$ as a Banach, weak symplectic manifold (a weak symplectic form is a closed 2-form that induces an injective map from the tangent to the cotangent bundle).
Fix a section $$\pi$$ in $$TM \otimes TM$$, and consider its induced bundle map $$\pi^{\sharp}: T^*M \to TM$$. For two given submanifolds $$C_{0}$$ and $$C_{1}$$ of $$M$$, denoted by $$\mathcal{C}_{\pi}(M; C_0,C_1)$$, the subpace of $$T^*PM$$ of “$$\pi$$-compatible path” from $$C_0$$ to $$C_1$$, and by $$p_i: \mathcal{C}_{\pi}(M; C_0,C_1) \to C_i$$, $$i=0,1$$, the canonical maps. Contrary to the case when $$C_0=C_1=M$$, $$\mathcal{C}_{\pi}(M; C_0,C_1)$$ is not in general a Banach submanifold. Although, $$\mathcal{C}_{\pi}(M; C_0,C_1)$$ is not a submanifold, one still can define the coisotropy criteria, using the notion of tangent bundle with fibers being the Zarisky tangent space at each point. Specifically, in the symplectic case, a subvariety (i.e., a common zero set of a family of smooth functions) is said to be coisotropic when its symplectic orthogonal bundle is contained in its tangent bundle.
The first main result of the paper says that $$\mathcal{C}_{\pi}(M; M,M)$$ is coisotropic in $$T^*PM$$ if and only if, $$\pi$$ is a Poisson bivector field. The later means that $$\pi$$ is a skew-symmetric 2-tensor field satisfying $$[\pi,\pi]=0$$, where $$[\,,\,]$$ is the Schouten-Nijenhuis bracket. The second main result asserts that, if $$\pi$$ is a Poisson bivector field, then $$\mathcal{C}_{\pi}(M; C_0,C_1)$$ is coisotropic in $$T^*PM$$ if and only if the subsets $$\mathrm{Im}(p_0)$$ and $$\mathrm{Im}(p_1)$$ are coisotropic in $$M$$ relative to $$C_0$$ and $$C_1$$ respectively, i.e., $$\pi^{\sharp}(N^*_xC_i) \subseteq T_xC_i$$, for all $$x \in \mathrm{Im}(p_i)$$, $$i=0,1$$.

##### MSC:
 53D17 Poisson manifolds; Poisson groupoids and algebroids 81T45 Topological field theories in quantum mechanics 53D20 Momentum maps; symplectic reduction 58H05 Pseudogroups and differentiable groupoids
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