A note on the linearity of real-valued functions with respect to suitable metrics (Q1768253)

The author proves that for every real-valued Morse function \(f\) on a smooth manifold and every neighborhood \(U\) of its critical points, a suitable Riemannian metric exists such that \(f\) is linear outside \(U\). The proof, based on the notion of flow set, relies on vertical extension to the flow set.