Certain conditions for a Finsler manifold to be isometric with a Finsler sphere (Q2087474)

The authors state that a Finsler manifold \((M,F)\) is diffeomorphic to the \(n\)-dimensional sphere whenever there exists a smooth function on \(M\) satisfying \(\nabla^2f=-kfg_{\nabla_f}\) for a positive constant \(k\). Here \(g_{\nabla_f}=g_{ij}(x,\nabla f)dx^i\otimes dx^j\), where\(g_{ij}=\frac{1}{2}[F^2]_{y^iy^j}\) and \((x^i,y^i)\) are the local coordinates on the tangent bundle of \(M\).