Two new families of Finsler connections on even-dimensional manifolds (Q2820179)

A pseudo-Finsler manifold is a triple \(F^n=(M,M',F^*)\), where \(M\) is a real \(n\)-dimensional smooth manifold, \(M'\) is a nonempty open submanifold of \(TM\) such that \(\pi(M')=M\), \(M'\cap\theta(M)=\emptyset\), with \(\theta\) the zero section of \(\pi:TM\longrightarrow M\), and \(F^*: M'\longrightarrow \mathbb{R}\) is a pseudo-Finsler function on \(M'\). An almost hypercomplex structure on an even-dimensional manifold consists of three globally defined almost complex structures \(J_i; i = 1, 2, 3\) satisfying the identities \(J_1J_2=-J_2J_1=J_3\). In the present paper, the author considers an even dimensional pseudo-Finsler manifold \(M^{2n}=(M,M',F^*)\) equipped with a nonlinear connection \(HM'\) on \(M'\). Using \(HM'\), he constructs an almost hypercomplex structure on \(M'\) and using this almost hypercomplex structure, he defines two new families of Finsler connections. Also, the author shows that to each Finsler connection \(\nabla\) is associated a linear connection \(D\) such that the almost hypercomplex structure is parallel with respect to \(D\).