  For a complete Riemannian manifold $M$ with an (1,1)-elliptic Codazzi
self-adjoint tensor field $A$ on it, we use the divergence type operator
${L_A}(u): = div(A\nabla u)$ and an extension of the Ricci tensor to extend
some major comparison theorems in Riemannian geometry. In fact we extend
theorems like mean curvature comparison theorem, Bishop-Gromov volume
comparison theorem, Cheeger-Gromoll splitting theorem and some of their famous
topological consequences. Also we get an upper bound for the end of manifolds
by restrictions on the extended Ricci tensor. The results can be applicable for
some kind of Riemannian hypersurfaces when the ambient manifold is Riemannian
or Lorentzian with constant sectional curvature.
