Two definitions of the injectivity radius For a Riemannian manifold $M$, usually, the injectivity radius of $p \in M$ is defined by the supremum of positive numbers $r$ such that $\exp_p|_{B(0, r)}$ is a diffeomorphism onto the image. Here, $B(0,r)$ is the geodesic ball of radius $r$. However, in Klingenberg's book, the injectivity radius is defined by by the supremum of positive numbers $r$ such that $\exp_p|_{B(0, r)}$ is injective. Are these two definitions equivalent?
Considering the global rank theorem, I tried to prove the exponential map at $p$ is of constant rank, but it seems that this try does not work. In fact, Klingenberg frequently says that the exponential map is just injective instead of saying the map is a diffeomorphism. To guarantee that the exponential map at $p$ is a diffeomorphism on a geodesic ball, is it enough to show that it is injective?
Thanks!

Edit: According to this answer, Klingenberg's Riemannian Geometry shows that $\exp: TM \to M$ is non-injective in an neighborhood of any critical point in Chapter 2.1, "Completeness and Cut Locus:"

The following result is from Klingenberg's Riemannian geometry book (Theorem 2.1.12):
Theorem. On a complete Riemannian manifold $(M,g)$, if $\mathrm{ker}(d\exp_p\vert_v)\neq 0$ for some $(p,v)\in TM$, then $\exp_p$ fails to be injective in every neighbourhood of $v$.
(Thus if injectivity fails infinitesimally, it also fails in every neighbourhood. This is of course false for a general smooth map.)

That being proven, Klingenberg is then free to give a weaker-looking, yet equivalent, alternative definition of injectivity radius, and apply it for the remainder of the text.
