On Conditions for Unrectifiability of a Metric Space

Abstract

We find necessary and sufficient conditions for a Lipschitz map f : E ⊂ ℝk → X into a metric space to satisfy ℋk(f(E)) = 0. An interesting feature of our approach is that despite the fact that we are dealing with arbitrary metric spaces, we employ a variant of the classical implicit function theorem. Applications include pure unrectifiability of the Heisenberg groups.

Keywords

• geometric measure theory
• unrectifiability
• metric spaces
• sard theorem
• carnot-carathéodory spaces
• 49q15
• 53c17