I need a reference to some basic facts about Morse theory on manifolds with boundary. Namely, if a critical point lies on the boundary, then the gradient of function might be nonzero and it brings some extra problems.

In fact my manifold is a domain in Euclidean space with smooth boundary, and I can assume that the function is linear. More precisely, I need a reference that could be used in the proof of 4.2.6. here.

**Postscript.** Let me thank Ryan Budney for the reference to "Morse lemmas for..." by Sergei Vakhrameev; it contains the needed lemmas for manifolds with corners (in particular for manifolds with boundary).

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