A couple of questions occured to me when reading about the Poincare-Hopf theorem. Any pointers or comments would be appreciated!
Let $M$ be a closed oriented Riemannian manifold and $V$ a vector field on $M$.
1) Is the statement that the zeros of $V$ are non-degenerate (by which I mean the Hessian at these points is invertible) equivalent to saying that $V$ is a transverse section of $TM$ (in the sense that it is transverse to the $0$ section)?
2) It is claimed that locally around each critical point $p$, there exist coordinates $y = (y_1,...,y_n)$ such that $V$ can be written as $V = y A_p$ for some constant matrix $A_p$. This sounds like something to do with the inverse function theorem, but how does one go about choosing the coordinates $y$ and the matrix $A_p$?
Thanks.
I've since heard back from the author of the book, who agrees that there is an issue with 2). It seems to me that the Hartman-Grobman theorem would suffice provided the zeros are hyperbolic. It was suggested to me that $V$ can be perturbed a little bit to make this true without changing the index of its zeros (since we want to prove the Poincare-Hopf theorem). How this is done, however, is a mystery to me.