  The aim of this article is to present a comparative review of Riemannian and
Finsler geometry. The structures of cut and conjugate loci on Riemannian
manifolds have been discussed by many geometers including H. Busemann, M.
Berger and W. Klingenberg. The key point in the study of Finsler manifolds is
the non-symmetric property of its distance functions. We discuss fundamental
results on the cut and conjugate loci of Finsler manifolds and note the
differences between Riemannian and Finsler manifolds in these respects. The
topological and differential structures on Riemannian manifolds, in the
presence of convex functions, has been an active field of research in the
second half of $20^{\text{th}}$ century. We discuss some results on Riemannian
manifolds with convex functions and their recently proved analogues in the
field of Finsler manifolds.
