Sasakian Finsler manifolds (Q2839036)

Let \(\pi :V\rightarrow M\) be a vector bundle with \(\dim M=n\) and \(\dim V=n+m\). Suppose \(\pi \) is endowed with a nonlinear connection \(N\) with the local coefficients \(N^a_i(x, y)\) for \(1\leq i\leq n\) and \(1\leq a\leq m\). This \(N\) yields a splitting of all geometric objects of \(V\) into vertical and horizontal part.NEWLINENEWLINEThe authors claim to give a generalization of almost contact (metric) geometry by dealing with this decomposition. Unfortunately, there are only two examples, both of them not related to the Sasakian case although the title and the abstract of the paper contain this word. The first example deals with \(M=\mathbb{R}^5\) and \(V=TM\) but without a specification for \(N\). In the second example \(M=\mathbb{R}^3\), \(V=TM\) and again \(N\) is not given although the authors claim that \(d(-y^2\delta y^1)=\delta y^1\wedge \delta y^2\) where \(\delta y^a=dy^a+N^a_idx^i\). There are also a lot of other errors: 1) in computing the so-called vertical and horizontal Ricci tensors the dimension of \(M\) is \(2n+1\) and that of \(V\) is \(4n+2\) without explications; 2) in all main computations an arbitrary torsion-free Finsler connection is used having no relationship with the Levi-Civita connection associated to the metric provided by the Sasaki structure.