We know from Morse theory that smooth manifold(with or without boundary) is a handlebody. However, I found a paper "Three-dimensional manifolds with boundary of nonnegative Ricci curvature" by Ananov, N. G.(2-AOS2); Burago, Yu. D.(2-AOS2); Zalgaller, V. A.(2-AOS2). They proved
Every 3-dimensional compact Riemannian manifold M (∂M≠∅) with nonnegative (at all points and in all directions) Ricci curvature (Ric≥0) and nonnegative mean curvature of the boundary (H≥0), which is positive at least at one point of ∂M, is diffeomorphic to a handle-body.
Why this need a proof? I am pretty sure that I missed something, but what is it? One thing I know is they proved the boundary is connected, which can be viewed as a splitting theorem.