Let $(M,g)$ be an open (:=complete, non-compact) Riemannian manifold with bounded geometry, in the sense that in some atlas of charts of radius $r_0>0$, the metric and all its derivatives are uniformly bounded (see, e.g. Cheeger--Gromov--Taylor). For simplicity, we only consider $\partial M=\emptyset$. My question is:
$\bullet$ For some $p\in M$ and a large number $R>0$, can we deduce that the geodesic ball $B(p,R)\subset M$ is a manifold-with-boundary of bounded geometry?
The boundedness of geometry for a manifold-with-boundary $\mathscr{N}$ is defined in the sense of T. Schick (https://arxiv.org/abs/math/0001108). It means that (1), the interior of $\mathscr{N}$ is of bounded geometry in the aforementioned sense; (2), $\partial \mathscr{N}$ can be flowed for a positive definite time along the inward unit normal; and (3), the second fundamental form of $\partial \mathscr{N}$ and all its derivatives are uniformly bounded, and the injectivity radius of $\partial \mathscr{N} \geq \iota_0 >0$.
I think this should be true, but I find it difficult to write down a prove. In particular, technicality arises if the geodesic sphere $\partial B(p,R)$ goes beyond the conjugate points of $p$.
A more general (but vague) question is:
$\bullet$ Let $p,R,M$ be as in the previous question. What can be said about the geometry of $\partial B(p,R)$?
Many Thanks!