Let $M$ be a smooth Riemannian manifold and let $p \in M$. Suppose $r \ll \text{inj}(p)$ (the injectivity radius at $p$) and fix $t \in (0,r)$ then define the map $$ T_pM \ni v \mapsto \exp_p(tv) \in M$$ I know this map is a diffeomorphism. How can I estimate its Lipschitz constant?
If I take $v_1, v_2 \in T_pM$ and I consider the geodesics $\gamma_1$ and $\gamma_2$ starting at $p$ with initial speed $v_1$ and $v_2$ respectively, it would be enough to prove $$ \frac{d}{dt} d \left(\gamma_1(t), \gamma_2(t)\right) \le |v_2-v_1| $$ to conclude that the Lipschitz constant is bounded by $t$. But even if I consider the differential of the distance given by $$ \frac{d}{dt} d (\gamma_1(t), \gamma_2(t))= \langle \dot{\alpha_t}(d_0), \dot{\gamma_2}(t) \rangle - \langle \dot{\alpha_t}(0), \dot{\gamma_1}(t) \rangle $$ where $d_0 = (\gamma_1(t), \gamma_2(t))$ and $\alpha_t$ is a unit speed geodesic connecting $\gamma_1(t)$ to $\gamma_2(t)$, I only obtain \begin{align*} \frac{d}{dt} d (\gamma_1(t), \gamma_2(t)) & = \langle \mathcal{P}^{\gamma_2}_{t \to 0}[\dot{\alpha_t}(d_0)], v_2 \rangle - \langle \mathcal{P}^{\gamma_1}_{t \to 0}[\dot{\alpha_t}(0)], v_1 \rangle \\ & = \langle \mathcal{P}^{\gamma_2}_{t \to 0}[\dot{\alpha_t}(d_0)] , v_2 \rangle -\langle \mathcal{P}^{\gamma_2}_{t \to 0}[\dot{\alpha_t}(d_0)], v_1 \rangle + \langle \mathcal{P}^{\gamma_2}_{t \to 0}[\dot{\alpha_t}(d_0)]-\mathcal{P}^{\gamma_1}_{t \to 0}[\dot{\alpha_t}(0)], v_1 \rangle \\ & \le |v_2-v_1| + \langle \mathcal{P}^{\gamma_2}_{t \to 0}[\dot{\alpha_t}(d_0)]-\mathcal{P}^{\gamma_1}_{t \to 0}[\dot{\alpha_t}(0)], v_1 \rangle \end{align*} where $\mathcal{P}^{\gamma}_{t_0 \to t_1}[v]$ is the parallel transport of $v \in T_{\gamma(t_0)}$from $\gamma(t_0)$ to $\gamma(t_1)$ along $\gamma$.
It is clear to me that the second summand must go to $0$ as $t \to 0^+$ and this would imply the result for small $t$. But I have not idea of how to prove that!
