On submaximal dimension of the group of almost isometries of Finsler metrics (Q2786833)

Let \((M,F)\) be a connected Finsler manifold where the Finsler metric \(F\) is strictly convex but not reversible. A diffeomorphism \(\phi: U\longrightarrow V\), with \(U,V\subset M\), is called an almost isometry if \(T(p,q, r) = T(\phi(p), \phi(q), \phi(r))\) for all \(p,q,r\in U\), where \(T\) is the ``triangular'' function given by \(T(p,q,r) := d(p,q) + d(q,r) - d(p,r)\) and \(d\) is the (generally nonsymmetric) distance corresponding to \(F\). The author shows that the second greatest possible dimension of the group of almost isometries of a Finsler metric \(F\) is \(\frac{n^2-n}{2}+1\) for \(n = \dim(M)\neq 4\) and \(\frac{n^2-n}{2}+2=8\) for \(n = 4\). If the group of almost isometries of \(F\) is of dimension greater than \(\frac{n^2-n}{2}+1\), then \(F\) is Randers, i.e., \(F(x, y) =\sqrt{g_x(y, y)} +\tau(y)\). Moreover, if \(n\neq 4\), the Riemannian metric \(g\) has constant sectional curvature and if, in addition, \(n\neq 2\), the 1-form \(\tau\) is closed and the metric admits group of local isometries of the maximal dimension \(\frac{n(n+1)}{2}\). In the remaining dimensions 2 and 4, the author describes all examples of the corresponding Finsler metrics with group of almost isometries of dimension 3 and 8.