## Almost symplectic $$N$$-linear connections in the bundle of accelerations.(English)Zbl 0972.53020

In $$E= \text{Osc}^2M$$ the local coordinates are denoted by $$(x,y^{(1)}, y^{(2)})$$. The almost symplectic $$d$$-structure is a tensor field $$a_{ij}(x,y^{(1)}, y^{(2)})$$, where $$a_{ij} = -a_{ji}$$, $$\det (a_{ij}) \neq 0$$. An $$N$$-linear connection $$D\Gamma (N) = (L_j{}^i{}_k, C_{(\alpha)j}^i{}_k)$$ $$(\alpha = 1,2)$$ with the property $$a_{ij|k} = 0$$, $$a_{ij |k}^{(\alpha)} = 0$$ $$(\alpha = 1,2)$$ is said to be an almost symplectic $$N$$-linear connection. The Obata’s operators are formed using the tensor field $$a_{ij}$$, and the relations between these fields and the curvature tensors are given. The set of all almost symplectic $$N$$-linear connections and the group of its transformations are obtained. Some invariants of this transformation are determined.

### MSC:

 53C05 Connections (general theory) 53D15 Almost contact and almost symplectic manifolds
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