There are explicit perturbations of Morse-Bott functions $f:X\to\mathbb{R}$ used in the literature (ex: Austin-Braam, Banyaga-Hurtubise, Bourgeois) to help solve various problems (ex: building Morse homology, and constructing nondegenerate contact forms). Namely, Morse functions $f_i$ are chosen on the critical submanifolds $C_i\subset crit(f)$ and then $0<\epsilon<<1$ is chosen so that $f+\epsilon\sum_i\rho_if_i$ is Morse (whose critical points have become $crit(f_i)$ on each $C_i$). Here $\rho_i$ is a bump function which has support on a tubular neighborhood of $C_i$ and takes value 1 in a smaller tubular neighborhood.

I am questioning how "rigid" this approach is, i.e. if I can replace the above description with something similar, to achieve a Morse function whose critical points live within the original $\operatorname{crit}(f)$. From the description, the $f_i$ are extended to be constant in the normal direction of $C_i\subset X$ (before being cut-off to zero). **Is this "constancy condition" crucial?**

**Can I instead extend $f_i$ to a small tubular neighborhood in any fashion (there is at least one way by Whitney's extension theorem) and then cut it off to zero?** This extension would have no requirement to be constant in the normal directions to $C_i$, and I can still choose $\epsilon$ arbitrarily small and have the critical manifolds $C_i$ be perturbed into $crit(f_i)$. In other words, the original approach pulls back $f_i$ to the normal bundle $N_{C_i}$ of $C_i$ so that locally the perturbation is $(c,x)\mapsto f_i(c)$, and I'm looking for other functions $(c,x)\mapsto \tilde{f}_i(c,x)$ satisfying $\tilde{f}_i(c,0)=f_i(c)$.

The more I think about this though, the more I understand why the original description is used (so that $\nabla f\perp\nabla f_i$ away from $C_i$, helping to ensure no new critical points appear).

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