Comparison Geometry for an Extension of Ricci Tensor

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