Is the set $\bigsqcup_{p\in M} \{v\in T_pM: |v|_g< r_p\}$open in $TM$?(where $r_p$ the injectivity radius at $p$)

Question

Let $(M,g)$ is a Riemannian manifold. (1)If $D_p$ is the largest domain on which $\exp_p$ can be a diffeomorphism, then is the set $$D=\bigsqcup_{p\in M} D_p$$ open in $TM$? (2)Likewise, if we denote $r_p$ the injectivity radius at $p\in M$, then is the set $$D_1=\bigsqcup_{p\in M} \{v\in T_pM: |v|_g< r_p\}$$ open in $TM$?

Answer 1

If $M$ is complete, then $D_1$ is open, which follows essentially from the fact that $p \mapsto r_p$ is continuous. For instance, consider the map $f : TM \to TM$ given by $$ f : v \mapsto \frac{v}{r_p}, $$ where $p$ is the footpoint of $v$; $f$ is continuous, and your $D_1$ is the inverse image under $f$ of the open set $$ \lbrace v \in TM : ||v|| < 1 \rbrace. $$