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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?

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  • $\begingroup$ 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, $\endgroup$
    – Roland F
    Commented Dec 5, 2023 at 7:21
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    $\begingroup$ @RolandF I don't see how that's relevant either $\endgroup$
    – ElijahB
    Commented Dec 6, 2023 at 6:16

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