# Lipschitz constant of the exponential map

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!

• What is $\mathrm{inj}(x)$? Thanks Mar 28, 2019 at 17:43
• It is the infectivity radius at $x$. Basically I am only considering a neighborhood of $0_p \in T_pM$ where the exponential map is a diffeomorphism onto some open neighborhood of $p$. Mar 28, 2019 at 17:45
• Yes, as long as you assume an upper sectional curvature bound. This essentially is the content of Rauch Comparison Theorem. Mar 28, 2019 at 21:24