Distance function on Riemannian

186 6-4. In Chapter 2, we started with a Riemannian metric and used it to define the Riemannian distance function. This problem shows how to go back the other way: the distance function determines the Riemannian metric. Let (M, 9) be a connected Riemannian manifold. (a) Show that if y: (-8,8) M is any smooth curve, then ly' (0g = lim dg (y(0), y(t)) 10 t (b) Show that if g and are two Riemannian metrics on M such that dg (p,q) = dz(pq) for all p q M, then g=. 175 cm 6.29. Suppose (M.g) is a connected Riemannian manifold and SCM is Wd (1.5) Osuch that B(x) is a closed geodesic ball contained in U and grad Ist- on B.(x). Let e be a positive constant less than 88. By definition of ds (X.S), there curve a : 10,6] - M (which we may assume to be parametrized by arc length) such that a(0) = x,a(b) S, and b=2(a) , so dois an admissible curve from ale) to S. On the one hand, de la(e).S) s (c|:41) = 6-8 Osuch that B(x) is a closed geodesic ball contained in U and grad Ist- on B.(x). Let e be a positive constant less than 88. By definition of ds (X.S), there curve a : 10,6] - M (which we may assume to be parametrized by arc length) such that a(0) = x,a(b) S, and b=2(a) , so dois an admissible curve from ale) to S. On the one hand, de la(e).S) s (c|:41) = 6-8