Classification of complete Finsler manifolds through a second order differential equation (Q931153)

The main result of the paper can be formulated as follows: Let \((M,g)\) be a complete connected Finsler manifold of dimension \(\geq 2\) admitting a nonconstant solution \(\rho\) of \(\nabla^H\nabla^H\rho=\varphi g\), where \(\nabla^H\) is the horizontal part of the Cartan derivative and \(\varphi\) is a smooth function on \(M\). Let \(N\) be the number of critical points of \(\rho\). Then \(M\) is conformally diffeomorphic to {\parindent=6mm \begin{itemize}\item[(a)] the sphere \(S^n\) if \(N=2\); \item[(b)] the Euclidean space \(\mathbf{E}^n\) if \(N=1\); \item[(c)] the product manifold \(I\times\overline{M}\) if \(N=0\), where \(I\subset\mathbb R\) is an open interval and \(\overline{M}\) is an \((n-1)\)-dimensional complete Finsler manifold. \end{itemize}}