http://arxiv.org/abs/math/0001108 This paper by Schick is nicely written and contains coordinate-wise and coordinate-free definitions of bounded geometry for manifolds with boundary.
In particular, it says that in this case the condition of having injectivity radius bounded from below translates to the following conditions:
- Normal collar: there is $r_c>0$ such that the geodesic collar $\partial M \times [0,r_c] \to M$ sending $(x,t)$ to $K(\exp_x(tv_x))$ (where $v_x$ is the unit inward normal vector) is a diffeomorphism into its image.
- Positive injectivity radius in $\partial M$.
- Positive injectivity radius in $M\setminus K(\partial M \times [0,r_c])$.
Most theorems that assume that $M$ is a borderless manifold with positive injectivity radius should have an analogue for the case of manifolds with border satisfying the above conditions.
