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.


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