Extending Morse functions from codimension $\geq$ 2 submanifolds

Question

Given $M^m \subset X^n$ manifolds where $n-m \geq 2$ and given a Morse function $f : M \to \mathbb{R}$ can I always extend $f$ to a Morse function on all of $X$? Further, if $f$ has critical points with distinct critical values and of increasing index, can I always extend it to all of $X$ with a Morse function that also has these properties?

Answer 1

Fix a bundle metric $g$ on $\nu(M)$. I'll denote $\nu_t(M)=\{(x,v) \in \nu(M)|g(v,v) \leq t\}$. On $\nu_1(M)$, consider the function $F(x,v)= \phi(g(v,v))f(x)+g(v,v)$ where $\phi:[0,1]\to [0,1]$ is a smooth cutoff function locally constant at zero and one with $\phi(0)=1$ and $\phi(1)=0$. $F$ is clearly Morse on $\nu_1(M)$ and constant along $\partial(\nu_1(M))$.

As $X-\nu_1(M)$ is a manifold with boundary, there exists a Morse function on $X-\nu_1(M)$ which is constant along $\partial(X-\nu_1(M))=\partial(\nu_1(M))$. Extend this function across $\nu_1(M)$ using $F$ to get the desired Morse function.