Manifolds with boundary and of bounded geometry
  For non-compact manifolds with boundary we prove that bounded geometry defined by coordinate-free curvature bounds is equivalent to bounded geometry defined using bounds on the metric tensor in geodesic coordinates.   We produce a nice atlas with subordinate partition of unity on manifolds with boundary of bounded geometry, and we study the change of geodesic coordinate maps. 
Thomas Schick (Penn State and Uni Muenster)