Does the Morse-Bott index of a critical point depend on the choice of metric?

Question

By the Morse lemma there exists a coordinate chart $(x_1,...,x_n)$ in the neighbourhood of a critical point $p$ of a Morse function $f : M^n \to \mathbb{R}$ such that \begin{equation*} f(x) = f(p) - \sum_{i=1}^{k_p} x_i^2 + \sum_{i=k_p+1}^n x_i^2 \end{equation*} where $k_p$ is the Morse index of $p$. In this way it is simple to see that the Morse index is intrinsic to $f$ and does not depend on the choice of a Riemannian metric on $M^n$ (as the definition of a Morse index via the Hessian would seem to suggest). How can one see that the index of Morse-Bott function also does not depend on a choice of metric?

Answer 1

You may think about index of critical points of a Morse-Bott function $f:M\to\mathbb R$ as follows: let $x\in V\subset M$ be a critical point, where $V$ is (component of) critical manifold. Then the tangent space $T_x$ is a sum $TV\oplus N_+\oplus N_-$. Here $N_+$ is maximal subspaces of vectors in direction of that $f$ is increasing, and $N_-$ is maximal subspaces of vectors in direction of that $f$ is decreasing; thise three subspaces are disjoint and sum of their dimensions equals $n$. And then, for this picture you may choose metric in neighborhood of $x$ such that $TV$, $N_+$ and $N_-$ will be pairwise orthogonal.