Let $M$ be a compact manifold of dimension $n$. A smooth function $f:M \to \mathbb{R}$ is called Morse-Bott if the set critical points of $f$ is a disjoint union of compact submanifolds $C_1,\ldots,C_k$ and
(1) for each critical point $x\in C_i$ the Hessian of $f$ in the transversal direction to $C_i$ is non-degenerate.
A well known analogue of Morse lemma, see e.g. the paper by A. Banyaga and D. E. Hurtubise, claims that the above condition (1) is equivalent to the following one:
(2) Suppose $\dim C_i = k < n$. Then for every $x\in C_i$ there are local coordinates $(u_1,\ldots,u_k, v_1,\ldots,v_{\lambda}, v_{\lambda+1},\ldots, v_{n-k})$ such that $$ f(u_1,\ldots,u_k, v_1,\ldots,v_{\lambda}, v_{\lambda+1},\ldots, v_{n-k}) = -\sum_{i=1}^{\lambda} v_i^2 + \sum_{i=\lambda+1}^{n-k} v_i^2. $$
One can regard the above property (2) as follows. First recall that a regular neighborhood of a submanifold $C \subset M$ is a smooth retraction $p:E \to C$ of an open neighborhood $E$ of $C$ in $M$ together with a structure of a vector bundle for $p$.
Also, let $p:E\to C$ be a vector bundle, and $m\geq2$ be a natural number. Then a smooth function $g:E \to\mathbb{R}$ will be called $m$-homogeneous if $g(tx) = t^m g(x)$ for all $x\in E$ and $t\in [0;+\infty)$.
Now consider the trivial vector bundle $$p:\mathbb{R}^n\to\mathbb{R}^{k}, \qquad p(u_1,\ldots,u_k, v_1,\ldots,v_{n-k}) = (u_1,\ldots,u_{k}).$$ Then (2) can be reformulated as follows:
(3) for every point $x\in C_i$ there exists an embedding $q:\mathbb{R}^n\to M$ such that $q^{-1}(C_i) = \mathbb{R}^k\times 0^{n-k}$, and the composition $f\circ q:\mathbb{R}^n \to \mathbb{R}$ coincides near $\mathbb{R}^k\times 0^{n-k}$ with a $2$-homogeneous function.
I am interested in whether the following "global" analogue of (3) holds.
Let $f:M \to \mathbb{R}$ be a Morse-Bott function and $C$ be its connected critical submanifold. Is that true that then there exists a regular neighborhood $q:E \to C$ of $C$ in $M$ such that $f \circ q: E \to \mathbb{R}$ coincides near $C$ with some $2$-homogeneous function $g:E \to \mathbb{R}$?
I will be grateful for any information about that problem.
