The Cut and Conjugate Loci, and Synge’s Theorem

In this chapter, the following assumptions are made throughout: •The Finsler structure F is positively (but perhaps not absolutely) homogeneous of degree one. Consequently, the associated metric distance function d may not be symmetric.• The Finsler manifold (M, F) is forward geodesically com plete. That is, all geodesics, parametrized to have constant Finslerian speed, are indefinitely forward extendible. By the Hopf-Rinow theorem, this is equivalent to saying that all forward Cauchy se quences are convergent. See Theorem 6.6.1 and §6.2D. Also, accord ing to Exercise 6.6.4, the completeness hypothesis is automatically satisfied whenever M is compact.• Unless we specify otherwise, all geodesics are parametrized to have unit Finslerian speed.