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Consider the Lagrange multiplier $$F(x, t, c) = f(x) + t( g(x) - c).$$ If $g$ satisfies your condition and $f$ is generic, then one can show that $$S:= \{ (x, t, c)\ |\ \nabla f(x) + t \nabla g(x) = 0 …

Maybe some of the following statements are not precise. Please correct them. Let $M$ be a compact smooth manifold. Let $f: M \to {\mathbb R}$ be a Morse function. Then a generic Riemannian metric $g$ …

Suppose we have an analytic function $f: U \to {\mathbb R}$, where $U\subset {\mathbb R}^n$ is an open subset and $0 \in U$ is a critical point of $f$. Thom conjectured that if a trajectory $x(t)$ of …

Let's start with a product manifold $M\times N$, with a product Riemannian metric $g_M\oplus g_N$. Consider a generic Morse function $f(x, y)$ on $M\times N$. Then its critical points are discrete and …