The Index Form and the Bonnet—Myers Theorem

Let (M, F) be a Finsler manifold, where F is C∞ on TM \0 and is positively (but perhaps not absolutely) homogeneous of degree 1. Fix T ∈ T _p M. Consider the constant speed geodesic σ(t) = exp _p (tT), 0 ⩽ t ⩽ r that emanates from p = σ(0) and terminates at q = σ(r). If there is no confusion, label its velocity field by T also. Let D _T denote covariant differentiation along σ, with reference vector T. This concept was introduced in the Exercise portion of 5.2.