Comparison geometry for an extension of Ricci tensor

2018/11/27 by Seyed Hamed Fatemi, Fatemi, S. H., Shahroud Azami +1 · 1 citation
Mathematics · Physics and Astronomy · #53C20 #53C23 #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.1811.11574

openalex publication_date 2018/11/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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∇ 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.

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