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I'm looking for a reference for the following result:

Let $X$ be a vector field on a manifold $M$ and let $C$ be a submanifold of $M$ formed by critical points of $X$. Assume that the Hessian of $X$ along $C$ has negative eigenvalues along the normal bundle of $C$. Then the vector field contracts a tubular neighborhood of $C$ to $C$.

The references that I managed to find always start by "Let $f:M \to \mathbb{R}$ be a Morse-Bott function..." I'm just looking for a reference that analyzes this type of results starting directly from a vector field that might not come from a potential.

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    $\begingroup$ Are you using "Hessian" in a non-standard way? It looks like you mean the derivative of the vector field along $C$, thinking of it as an endomorphism of the normal bundle, i.e. if the vector field were a gradient, this would be the Hessian of the potential function. $\endgroup$ Commented May 17, 2023 at 18:03
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    $\begingroup$ I don't know if your theorem has been written up, but it is a standard argument using the length functional in a normal bundle, i.e. your vector field is "inward pointing" and the distance from C is strictly decreasing along the flow, at least near-enough to C. $\endgroup$ Commented May 17, 2023 at 18:09
  • $\begingroup$ @RyanBudney yes, I’m using Hessian as the derivative of the vector field. I’m pretty sure I could furnish a proof myself. But since this must be “standard” I thought it might be written somewhere. I don’t think it adds much to the paper to prove it myself and I don’t want to use some magical sentence like “it’s known…”. But if there does note exist any reference, I will give a short proof I guess $\endgroup$ Commented May 18, 2023 at 19:27
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    $\begingroup$ There's quite a few results like these that tend to appear multiple times in various papers whose primary focus is some other topic. Since there isn't a standard reference it can be difficult to keep track of where these results appear. Basic results about Morse functions on various families of stratified spaces (manifolds with boundary, manifolds with cubical corners, ... ) are also like this. Probably the best strategy is to include a proof in your paper, and if you are lucky a reader or your referee might be able to locate a reference. That said, perhaps check Bott's papers first? $\endgroup$ Commented May 18, 2023 at 21:34
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    $\begingroup$ Another thing you can do is use MathSciNet to check the titles of papers that cite Bott's Morse Theory papers. There will almost certainly be a large number of them, but perhaps the titles will give some clues as to which papers are more likely to contain the result you are looking for. $\endgroup$ Commented May 18, 2023 at 21:41

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