Sánchez González, Luis FranciscoJiménez Sevilla, María Del Mar2023-06-202023-06-202011-070362-546X10.1016/j.na.2011.03.004https://hdl.handle.net/20.500.14352/42295The authors wish to thank Jesús Jaramillo for many helpful discussions. Supported in part by DGES (Spain) Project MTM2009-07848. L. Sánchez-González has also been supported by grant MEC AP2007-00868.Let us consider a Riemannian manifold M (either separable or non-separable). We prove that, for every ε>0, every Lipschitz function f:M→R can be uniformly approximated by a Lipschitz, C1-smooth function g with . As a consequence, every Riemannian manifold is uniformly bumpable. These results extend to the non-separable setting those given in [1] for separable Riemannian manifolds. The results are presented in the context of Cℓ Finsler manifolds modeled on Banach spaces. Sufficient conditions are given on the Finsler manifold M (and the Banach space X where M is modeled), so that every Lipschitz function f:M→R can be uniformly approximated by a Lipschitz, Ck-smooth function g with (for some C depending only on X). Some applications of these results are also given as well as a characterization, on the separable case, of the class of Cℓ Finsler manifolds satisfying the above property of approximation. Finally, we give sufficient conditions on the C1 Finsler manifold M and X, to ensure the existence of Lipschitz and C1-smooth extensions of every real-valued function f defined on a submanifold N of M provided f is C1-smooth on N and Lipschitz with the metric induced by M.engjournal articlehttp://www.sciencedirect.com/science/article/pii/S0362546X11001106http://www.sciencedirect.com/restricted access517.988Riemannian manifoldsFinsler manifoldsGeometry of Banach spacesSmooth approximation of Lipschitz functionsSmooth extension of Lipschitz functionsAnálisis funcional y teoría de operadores