On some problems on smooth approximation and smooth extension of Lipschitz functions on Banach-Finsler Manifolds

Let us consider a Riemannian manifold $M$ (either separable or non-separable). We prove that, for every $\epsilon>0$, every Lipschitz function $f:M\rightarrow\mathbb R$ can be uniformly approximated by a Lipschitz, $C^1$-smooth function $g$ with $\Lip(g)\le \Lip(f)+\epsilon$. As a consequence, every Riemannian manifold is uniformly bumpable. The results are presented in the context of $C^\ell$ 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 $f:M\rightarrow \mathbb R$ can be uniformly approximated by a Lipschitz, $C^k$-smooth function $g$ with $\Lip(g)\le C \Lip(f)$ (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 $C^\ell$ Finsler manifolds satisfying the above property of approximation. Finally, we give sufficient conditions on the $C^1$ Finsler manifold $M$ and $X$, to ensure the existence of Lipschitz and $C^1$-smooth extensions of every real-valued function $f$ defined on a submanifold $N$ of $M$ provided $f$ is $C^1$-smooth on $N$ and Lipschitz with the metric induced by $M$.