Math 6510 • Differentiable Manifolds

Let \(f\) be a smooth function on a smooth manifold \(M^n\). A critical point \(x_0\in M\) is nondegenerate if in some coordinate system the Hessian matrix of second partials \[ \frac {\partial ^2 f}{\partial x^i\partial x^j}(x_0) \] is nonsingular at \(x_0\). A function all of whose critical points are nondegenenerate is called a Morse function. Show that the critical points of a Morse function are isolated. Given any smooth function \(f\in C^\infty (M)\), show that for every \(\epsilon >0\) there is a Morse function \(g\in C^\infty (M)\) with \[ \sup _{x\in M}|f(x)-g(x)|<\epsilon . \]