  We introduce some new curvature quantities such as conformal Ricci curvature
and bi-Ricci curvature and extend the classical Myers theorem under these new
curvature conditions. Moreover, we are able to obtain the Myers type theorem
for minimal submanifolds in ambient manifolds with positive bi-Ricci curvature.
Some topological applications are discussed. We also give examples of manifolds
of positive bi-Ricci curvature and prove that the connect sum of manifolds of
positive bi-Ricci curvature admits metrics of positive bi-Ricci curvature.
