Smooth approximation of lipschitz functions on finsler manifolds

We study the smooth approximation of lipschitz functions on finsler manifolds,keeping control on the corresponding lipschitz constants. we prove that,given a lipschitz function f: m → r defined on a connected,second countable finsler manifold m,for each positive continuous function ε: m → (0,∞) and each r > 0,there exists a c 1 -smooth lipschitz function g: m → r such that | f (x) - g (x) | ≤ ε (x),for every x ∈ m,and l i p (g) ≤ l i p (f) + r. as a consequence,we derive a completeness criterium in the class of what we call quasi-reversible finsler manifolds. finally,considering the normed algebra c b 1 (m) of all c 1 functions with bounded derivative on a complete quasi-reversible finsler manifold m,we obtain a characterization of algebra isomorphisms t: c b 1 (n) → c b 1 (m) as composition operators. from this we obtain a variant of myers-nakai theorem in the context of complete reversible finsler manifolds. © 2013 m. i. garrido et al.