# Geodesic and Riemannian distances are equivalent via exponential map

Exercise 5.40, page 134, in the book A Course in Differential Geometry (Graduate Studies in Mathematics) by Thierry Aubin asks to prove that given $$X, Y \in \mathbb R^n$$, for any $$\varepsilon>0$$, there is some $$\eta>0$$ such that if $$0\leq t \leq \eta$$, then $$d(\exp_P(tX), \exp_P(tY)) \leq (1+\varepsilon)\|X-Y\|t,$$ where $$(M,g)$$ is a smooth Riemannian manifold of dimension $$n$$ and $$P \in M$$ is fixed. I'm wonder if the constant $$\eta$$ can be chosen independent of $$X$$ and $$Y$$. Furthermore, is there any similar estimate, however, in the opposite direction, namely $$(1-\varepsilon)\|X-Y\|t \leq d(\exp_P(tX), \exp_P(tY)).$$ Thank you so very much.