Comparison geometry for an extension of Ricci tensor

TL;DR

This paper extends classical comparison theorems in Riemannian geometry using an extended Ricci tensor and a divergence operator associated with a Codazzi tensor, broadening their applicability to new geometric contexts.

Contribution

It introduces an extension of the Ricci tensor and divergence operator to generalize key comparison theorems and topological results in Riemannian geometry.

Findings

Extended mean curvature comparison theorem

Generalized Bishop-Gromov volume comparison theorem

New bounds for manifold ends based on extended Ricci tensor

Abstract

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.