Cut Loci and Distance Functions

f 0 q ( ) = cos ; where denotes the inm um of angles between and the initial directions of minimal geodesics from q to p. The behavior of the distance function dp is closely related to the structure of the cut locus of p. Recall that the cut locus C(p) of p 2 M is given as follows: for any unit speed geodesic u emanating from p with the initial direction u = _ u(0) 2 UpM, there exists the last parameter value ip(u) up to which u is a minimal geodesic segment, namely uj[0; t] realizes the distance d(p; u(t)) for 0 < t ip(u). We call u(ip(u)) the cut point of p and ip(u) the cut distance to p along u. Then the cut locus C(p) of p is dened as