Let $\tilde M$ be a compact Riemannian manifold, without conjugate points. Denote by $M$ its universal cover. Then in this paper, it is claimed that every geodesic is globally length minimizing.
Why is this true?
Is it true for any simply connected manifold without conjugate points, or the fact $M$ is a universal cover of a compact manifold plays a role?
Note: If $M$ is not simply connected, than the lack of conjugate points does not imply geodesics are minimizing. (For example look at the cylinder).
The fact that it is the universal cover of a compact manifold is not essential, what you need is completeness of the Riemannian manifold $\tilde M$ (the universal cover of a compact manifold is always complete) and its simple connectivity. Assuming these two properties and the NCP property, just follow the proof of the Cartan-Hadamard theorem to conclude that for each $p\in \tilde M$ the exponential map $\exp_p: T_p \tilde M\to \tilde M$ is a diffeomorphism. From this, you see that any two points are connected by a unique geodesic. This implies that geodesics are global length minimizers.