Deicke's theorem

Statement

Let $(M,F)$ be a Finsler manifold. Let $x\in M$ be a point. The norm $F_x$ in the tangent space at $x$ is a Euclidean norm if and only if the mean Cartan torsion is 0 for all tangent vectors at $x$.

Note that for a Riemannian metric, the norm is Euclidean on each tangent space, and the mean Cartan torsion is identically zero.