Deicke's theorem

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Template:Finsler metric result

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.