On pages no. 52-56 of "Lectures on Morse Homology" by Augustin Banyaga and David Hurtubise presented a proof of Morse Lemma using homotopy method AKA Palais proof using Moser "path method". There are several statements I did not understand while trying to read the complete proof.
The proof starts like this: "By replacing f by f-f(p) and by choosing a suitable coordinate chart on M we may assume that the function f is defined on a convex neighborhood $U_0$ of $0\in R^m$ where $f(0)=0, df(0)=0$ and the matrix of the Hessian at $0\in R^m$, $M_0(f) = A = (\frac{\partial^2 f}{\partial x_i \partial x_j}(0))$, is a diagonal matrix with the first k diagonal entries equal to -1 and the rest equal to +1.
The matrix A induces a function $\widetilde{A}(x) = x^tAx=<Ax,x>=\sum_{j=1}^{m}\delta_j x_j^2$ where $\delta_j = \frac{\partial^2 f}{\partial^2 x_j}(0) = \pm1$ for all j=1,...,m. We want to prove that there are neighborhoods U and U' of $0$ with $U \subseteq U_0$ and a diffeomorphism $\varphi: U->U'$ such that:
$f\circ\varphi = \widetilde{A}$. (3.1)
The Idea of the path method is to interpolate f and $\widetilde {A}$ by a path such as, $f_t = \widetilde{A} + t(f-\widetilde{A})$ (3.2)
and to look for a smooth family $\varphi_t$ of diffeomorphisms such that $f_t\circ\varphi_t=f_0=\widetilde{A}$. (3.3)
Then $\varphi=\varphi_1$ will satisfy $f\circ\varphi=\widetilde{A}$.
We get $\varphi_t$ as a solution of the differential equations: $\frac{d\varphi_t}{dt}(x)=\xi_t(\varphi_t(x)); \varphi_0(x)=x$ where the smooth family $\xi_t$ is the tangent along the curves $t\mapsto\varphi_t(x)$. Taking the partial derivative with repect to t of both sides of (3.3) gives
$(\dot{f_t}\circ\varphi_t + (\xi_t\cdot f_t)\circ\varphi_t)(x)=0$ (3.4) for all $x \in U$ where $\dot {f_t}$ denotes $\frac{\partial f_t}{\partial t}$.
Thus, $(\dot{f_t} + \xi_t \cdot f_t)(y) = 0$ (3.5) for all $y \in U'$.
But $\dot{f_t} = f - \widetilde{A}$ by (3.2), and therefore (3.5) becomes $df_t(\xi_t)=g$ (3.6) where $g=\widetilde{A} - f$."
My questions regarding this part of the proof are these:
First, "Taking the partial derivative with respect to t of both sides of (3.3) gives (3.4)" - why is that so? I tried defining a function $h:R^{n+1}\rightarrow R^n$ such that $h(t,x)=f_t\circ\varphi_t$ and differentiating it using the chain rule, and what i got is:
$\frac{\partial h}{\partial t}(t,\varphi_t)+(\xi_t \circ \varphi_t) \cdot \frac{\partial h}{\partial x}(t,\varphi_t)$
and not:
$\frac{\partial h}{\partial t}(t,\varphi_t)+(\xi_t \circ \varphi_t) \cdot h(t,\varphi_t)$
as it seems I should get by (3.4).
Second, "and therefore (3.5) becomes $df_t(\xi_t)=g$" - to my understanding (3.5) should become: $\xi_t \cdot f_t=g$, meaning that $df_t(\xi_t)=\xi_t \cdot f_t$ - why is this the case here?
Moreover, the proof ends like this (there is a middle part to the proof which I omitted because I understood it...): "$B_0^t$ (which is defined by $\frac{\partial^2 f_t}{\partial x_i \partial x_j})(0)$) is non-degenerate for all $0\leq t\leq 1$, and there exists a neighborhood $\widetilde{U}$ of $0 \in R^m$ such that $B_x^t$ is also non-degenerate for all t. For $x \in U$, we have a unique solution $\xi_t$ of $<B_x^t \xi_t,x>=<G_x x,x>$ and this solution depends smoothly on both x and t. That is, we have a smooth solution to (3.6) defined on $\widetilde{U}$. Clearly, $\xi_t(0)=0$ since $B_0^t$ is non-degenerate. Hence, by shrinking $\widetilde{U}$ we can integrate $\xi_t$ and get a smooth family of diffeomorphisms $\varphi_t$ from a smaller neighborhood $U$ of $0$ to another neighborhood $U'$ of $0$ which satisfies $f_t \circ \varphi_t = f_0 = \widetilde{A}$."
I also have two questions regarding this part of the proof:
It is said that "$B_0^t$ is non-degenerate for all $0\leq t\leq 1$, and there exists a neighborhood $\widetilde{U}$ of $0 \in R^m$ such that $B_x^t$ is also non-degenerate for all t". I don't understand why this statement is correct. What we know is that for every t between 0 and 1, there exists a neighborhood $\widetilde{U}_t$ of $0 \in R^m$ such that $B_x^t$ is also non-degenerate for all x in $\widetilde{U}_t$, but this statement says something stronger, which I do not really understand why this something is correct.
At the very last sentence of the proof it is said that "by shrinking $\widetilde{U}$ we can integrate $\xi_t$ and get a smooth family of diffeomorphisms $\varphi_t$ from a smaller neighborhood $U$ of $0$ to another neighborhood $U'$ of 0 which satisfies $f_t \circ \phi_t=f_0=\widetilde{A}$." - what does it mean "by shrinking"? Does it have anything to do with the neighborhood I asked about in my third question?