Manifolds with boundary and of bounded geometry

TL;DR

This paper establishes the equivalence of different definitions of bounded geometry on non-compact manifolds with boundary and constructs useful coordinate systems for such manifolds.

Contribution

It proves the equivalence of curvature-based and coordinate-based bounded geometry definitions and develops a suitable atlas with partitions of unity for manifolds with boundary.

Findings

Bounded geometry definitions are equivalent in this setting.

Constructed a well-behaved atlas with partitions of unity.

Analyzed the stability of geodesic coordinate maps.

Abstract

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.