Geometric transformation theorem, fundamental groups and monotone of numbers of almost Euclidean factors of geodesic balls

In \cite{HH}, H. this http URL introduced the generalized Reifenberg condition which describes the non-increasing property of numbers of almost Euclidean factors of geodesic balls and gave a transformation theorem under this condition.

In this note, we will prove a transformation theorem under a non-decreasing property compared with the non-increasing property above and give an example that transformation theorem is false without the monotone property.

By these transformation theorems, as the main results in \cite{H}, we will show that for an open manifold with nonnegative Ricci curvature, if its universal cover is polar at infinity and the number of almost Euclidean factors of geodesic balls in the universal cover is monotone, then its fundamental group is finitely generated and virtually abelian.