  We prove a new generalization of the Cheeger-Gromoll splitting theorem where
we obtain a warped product splitting under the existence of a line. The
curvature condition in our splitting is a curvature dimension inequality of the
form $CD(0,1)$. Even though we have to allow warping in our splitting, we are
able to recover topological applications. In particular, for a smooth compact
Riemannian manifold admitting a density which is $CD(0,1)$, we show that the
fundamental group of $M$ is the fundamental group of a compact manifold with
nonnegative sectional curvature. If the space is also locally homogeneous, we
obtain that the space also admits a metric of non-negative sectional curvature.
Both of these obstructions give many examples of Riemannian metrics which do
not admit any smooth density which is $CD(0,1)$.
