  We consider a condition on the Ricci curvature involving vector fields, which
is broader than the Bakry-\'Emery Ricci condition. Under this condition volume
comparison, Laplacian comparison, isoperimetric inequality and gradient bounds
are proven on the manifold.
  Specializing to the Bakry-\'Emery Ricci curvature condition, we initiate an
approach to work on the original manifold, which yields, under a weaker than
usual assumption, the results mentioned above for the {\it original manifold}.
These results are different from most well known ones in the literature where
the conclusions are made on the weighted manifold instead.
  Applications on convergence and degeneration of Riemannian metrics under this
curvature condition are given. To this effect, in particular for the
Bakry-\'Emery Ricci curvature condition, the gradient of the potential function
is allowed to have singularity of order close to $1$ while the traditional
method of weighted manifolds allows bounded gradient. This approach enables us
to extend some of the results in the papers \cite{Co}, \cite{ChCo2},
\cite{zZh}, \cite{TZ} and \cite{WZ}. The condition also covers general Ricci
solitons instead of just gradient Ricci solitons.
