A proof of the Morse-Bott Lemma

It is often said that the Morse-Bott Lemma can be viewed as a “parameterized” Morse Lemma, and its proof should follow from the differentiability of the methods used to prove the Morse Lemma. The goal of this expository paper is to fill in the details. We present Palaisʹ proof of the Morse Lemma using Moserʹs path method, which yields the necessary differentiability.