In the present paper, we define Morse--Bott functions on manifolds with boundary which are generalizations of Morse functions and show Morse--Bott inequalities for these manifolds.
"Morse--Bott Inequalities for Manifolds with Boundary." Tokyo J. Math. 41 (1) 113 - 130, June 2018. https://doi.org/10.3836/tjm/1502179256