Let $(W;L,L')$ be a Lagrangian cobordism with $L, L'$ closed and connected Lagrangians. We will show that under conditions on the fundamental group, sometimes a monotone Lagrangian cobordism is an h-cobordism and when $W$ is exact then it is diffeomorphic to $W\cong \mathbb{R}\times L$. Part of the result presented is joint work with Jean-François Barraud.