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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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