# Differential of exponential map on Cartan-Hadamard manifold

I'm currently learning about Cartan-Hadamard manifold (simply connected, complete Riemannian manifold with nonpositive sectional curvature), and there are three equivalent conditions describing Cartan-Hadamard manifold:

Let (M,g) be a simply connected, complete Riemannian manifold. TFAE:

(1) M has nonpositive sectional curvature.

(2) The differential of each exponential map $$exp_{p}:T_{p}M \rightarrow M$$ is length increasing: $$|(dexp_p)_{v}(\tilde{v})|\geq|\tilde{v}|$$ for any $$v,\tilde{v}\in T_pM, p\in M$$

(3) $$exp_p$$ is distance increasing: $$d(exp_p(v),exp_p(w))\geq |v-w|$$, for any $$v,w\in T_pM$$, $$p\in M$$.

I'm stuck on proving (2) $$\implies$$ (1).

My guess is that I should use some sort of index form argument to show (2) implies (1), but honestly I don't know where to get my hands on. Can anyone give me a hint on this?

• Simply the opposite of positive curvature: any n-1-sphere around any point has equatorial length greater than $2 \pi r$ in the local tangent map to $\mathbb R^n$ yielding the curvature by the limit the angular deficit divided by the area, Commented Dec 5, 2023 at 7:21
• @RolandF I don't see how that's relevant either Commented Dec 6, 2023 at 6:16