A result about C2-rectifiability of one-dimensional rectifiable sets: application to a class of one-dimensional integral currents

Let γ, τ : [a, b] → Rk+1 be a couple of Lipschitz maps such that γ’ = | γ’| τ almost everywhere in [a, b]. Then γ ([a, b]) is a C2-rectifiable set, namely it may be covered by countably many curves of class C2 embedded in Rk+1. As a consequence, projecting the rectifiable carrier of a one-dimensional generalized Gauss graph provides a C2-rectifiable set.