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?
You are on the right track but it seems you have misinterpreted the meaning of (2). I would suggest reading up on Meyer's theorem before continuing forward with this problem. Attached is context and reference from a paper on Hadamard Manifolds written by Japanese scholar Kiyoshi Shiga.
As the starting point in the study of Riemannian manifolds of non- positive curvature, we first recall the Cartan-Hadamard theorem
Theorem. Let H be an n-dimensional simply connected complete Riemannian manifold o f nonpositive curvature. Then H is diffeomorphic to the n-dimensional Euclidean space Rft. More precisely, at any point p E H, the exponential mapping expp: Hp----,,"H is a diffeomorphism.
This theorem presents a clear contrast to Meyer's theorem: if a complete Riemannian manifold M is of strictly positive Ricci curvature, i.e., Ricci curvature >k>O for some k, then M is com- pact.
A simply connected complete Riemannian manifold of nonpositive curvature is called a Hadamard manifold or a Cartan-Hadamard manifold after the Cartan-Hadamard theorem. Unless otherwise mentioned, Hwill always denote a Hadamard manifold throughout this report.
From the Cartan-Hadamard theorem, there follow several basic pro- perties of Riemannian manifolds of nonpositive curvature. For example, any pair of distinct points of a Hadamard manifold can be joined uniquely by a geodesic segment. It also follows that the fundamental group of a compact Riemannian manifold of nonpositive curvature is an infinite group.