Milnor fibration theorem for differentiable maps

In Cisneros-Molina et al. (São Paulo J Math Sci, 2023. https://doi.org/10.1007/s40863-023-00370-y) it was proved the existence of fibrations à la Milnor (in the tube and in the sphere) for real analytic maps \(f:({\mathbb {R}}^n,0) \rightarrow ({\mathbb {R}}^k,0)\) $f:({\mathbb{R}}^n,0)\to ({\mathbb{R}}^k,0)$ , where \(n\ge k\ge 2\) $n\ge k\ge 2$ , with non-isolated critical values. In the present article we extend the existence of the fibrations given in Cisneros-Molina et al. (São Paulo J Math Sci, 2023. https://doi.org/10.1007/s40863-023-00370-y) to differentiable maps of class \(C^{\ell }\) $C^{\ell }$ , \(\ell \ge 2\) $\ell \ge 2$ , with possibly non-isolated critical value. This is done using a version of Ehresmann fibration theorem for differentiable maps of class \(C^{\ell }\) $C^{\ell }$ between smooth manifolds, which is a generalization of the proof given by Wolf (Michigan Math J 11:65–70, 1964) of Ehresmann fibration theorem. We also present a detailed example of a non-analytic map which has the aforementioned fibrations.