A class of Finsler metrics with almost vanishing \(H\)-curvature (Q2834827)

In this paper, the author studies a class of Finsler metrics on \({\mathbb B}^n(\nu)\) with orthogonal invariance (spherically symmetry).NEWLINENEWLINELet \(F\) be a Finsler metric on \({\mathbb B}^n(\nu)\). The \(\Xi\)-curvature and the \(H\)-curvature of the metric \(F\) is defined by NEWLINE\[NEWLINE \Xi_j:=S_{\cdot j|i}y^i - S_{|j},\quad H_{ij} = \frac{1}{4}\left(\Xi_{i\cdot j}+ \Xi_{j\cdot i}\right), NEWLINE\]NEWLINE where \(S\) is the \(S\)-curvature defined from the distortion of \(F\), whereas \(_|\) and \(\cdot\) denote the horizontal and vertical covariant derivatives, respectively. We say that the Finsler metric \(F\) has almost vanishing \(H\)-curvature if \(H_{ij} = \frac{n+1}{2}\theta F_{y^i y^j} \), and has almost vanishing \(\Xi\)-curvature if \(\Xi_j = -(n+1)F^2 \left(\frac{\theta}{F}\right)_{y^j}\), where \(\theta\) is a \(1\)-form on \(M\) and \(n = \dim (M)\). It is well known that if \(F\) has almost vanishing \(\Xi\)-curvature, then it has also almost vanishing \(H\)-curvature, but not the converse in general. It is also known that a Finsler metric \(F\) on \({\mathbb B}^n(\nu)\) is orthogonal invariant if and only if there is a function \(\phi: [0, \nu) \times {\mathbb R} \to {\mathbb R}\) such that NEWLINE\[NEWLINE F(x, y) = |y|\phi\left(|x|, \frac{\langle x, y\rangle}{|y|}\right), NEWLINE\]NEWLINE where \((x, y) \in T{\mathbb B}^n(\nu)\setminus \{0\}\).NEWLINENEWLINEIn this paper, the author finds an equation that characterizes Finsler metrics of almost vanishing \(H\)-curvature, and as a consequence, he shows that all orthogonally invariant Finsler metrics of almost vanishing \(H\)-curvature are of almost vanishing \(\Xi\)-curvature and the corresponding \(1\)-forms are exact. This is a generalization of a result previously known only in the case of metrics with vanishing \(H\)-curvature.