A Metric Splitting of Alexandrov Spaces: Boundary Strata of nonnegatively curved Alexandrov Spaces and a Splitting Theorem

Wörner Andreas

Abstract

Alexandrov spaces are metric generalizations of Riemannian manifolds with sectional curvature bounds. The boundary of an Alexandrov space M may decompose into several boundary strata. If M has positive curvature, it is quite well understood how the number of boundary strata determines the homeomorphism type of M as a stratified space. This book deals with the case that M has nonnegative curvature. The author Andreas Wörner begins with an introduction to Alexandrov geometry, which requires only some familiarity with Riemannian geometry. Then boundary strata are investigated more closely. After all prerequisites are given, a splitting theorem is proved as the main result in this book. More precisely, let M be compact and of dimension n. Assume that M has k+1 boundary strata such that their common intersection is empty, but any intersection of k strata is nonempty. Then M is isometric to a metric product of Alexandrov spaces S and D, where S has dimension n-k and is isometric to each intersection of k boundary strata. It is remarkable that the theorem provides in general non-flat factors.