I would like to prove the following statement (from page 92 in Riemannian Geometry, An modern introduction by Issac Chavel).
Suppose that $M$ is a Riemannian manifold with the constant sectional curvarure $\kappa = 1$ and that there is a point $p\in M$, such that $\forall v \in cl(B(p,\pi)) \subset T_pM$, $\gamma_v(1)$ is well-defined (i.e., The exponential map is well-defined on that closed ball). Then $M$ is compact (hence complete) and the image of the sphere exp$_pS(p,\pi)$ is exactly 1 point.
If $M = \mathbb{S}^n$, then I can show this by direct computation. But without this, can anyone give me a hint or some material so that I can prove it ?.