TITLE: Hadamard theorem and length of geodesics
QUESTION [0 upvotes]: I read  sevral times  in articles that:
"By Hadamard theorem, there is no closed  geodesics of length $2\pi$ in a space of sectional curvature $<0$?
Can someone give me a reference of this theorem, because i don't find it !
Thank you in advance

REPLY [0 votes]: You can find a proof of the Cartan-Hadamard theorem for instance in do Carmo's book "Riemannian Geometry" or in Peterson's book with the same title. (Actually, most advanced RG textbooks will contain a proof.) Here is the full statement:
Theorem. Suppose that $M$ is a complete simply-connected Riemannian manifold of sectional curvature $\le 0$ everywhere. Then for each point $p\in M$ the exponential map $\exp_p: T_pM\to M$ is a diffeomorphism.
To conclude that such $M$ cannot contain a closed geodesic of positive length ($2\pi$ is irrelevant), note that each nonconstant geodesic in $M$ (regarded as a set!) has the form $\exp_p({\mathbb R} u)$ for some $p\in M$ and a unit tangent vector $u\in T_pM$. By the CH theorem,  $\exp_p({\mathbb R} u)$ is diffeomorphic to the real line, hence, cannot be a loop.
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