The Morse–Bott–Kirwan condition is local

Kirwan identified a condition on a smooth function under which the usual techniques of Morse–Bott theory can be applied to this function. We prove that if a function satisfies this condition locally then it also satisfies the condition globally. As an application, we use the local normal form theorem to recover Kirwan’s result that the norm square of a momentum map satisfies Kirwan’s condition.