Department of Pure Mathematics, Faculty of Science, Imam Khomeini International University, Qazvin, Iran.
10.22108/msci.2026.146917.1763
Abstract
In this paper, we study comparison theorems for a Riemannian manifold $M^n$ under a lower bound condition on the Ricci curvature involving vector fields and gradient vector fields. We first establish Laplacian and volume comparison theorems for Riemannian manifolds endowed with a modified Ricci curvature as follow
and then for manifolds satisfying the Bakry–Émery Ricci curvature condition as follow
\begin{equation}\nonumber
Ric+Hess h\geq (n-1)k,
\end{equation}
for some smooth function $h$. Furthermore, we show that these comparison theorems remain valid, in particular, for the shrinking, steady, and expanding gradient Ricci solitons under the non-collapsing volume condition, and even without this condition when the soliton potential function is bounded. Consequently, we extend all of these results to almost Ricci solitons and gradient almost Ricci solitons. By using the obtained comparison results, we also derive a segment inequality for Riemannian manifolds $M^n$ with bounded Ricci curvature.
Sohrabpour, M. , Hajiaghasi, S. and Azami, S. (2026). Comparison Theorems and Their Applications on Riemannian Manifolds with Ricci Curvature Bounds. Mathematics and Society, (), -. doi: 10.22108/msci.2026.146917.1763
MLA
Sohrabpour, M. , , Hajiaghasi, S. , and Azami, S. . "Comparison Theorems and Their Applications on Riemannian Manifolds with Ricci Curvature Bounds", Mathematics and Society, , , 2026, -. doi: 10.22108/msci.2026.146917.1763
HARVARD
Sohrabpour, M., Hajiaghasi, S., Azami, S. (2026). 'Comparison Theorems and Their Applications on Riemannian Manifolds with Ricci Curvature Bounds', Mathematics and Society, (), pp. -. doi: 10.22108/msci.2026.146917.1763
CHICAGO
M. Sohrabpour , S. Hajiaghasi and S. Azami, "Comparison Theorems and Their Applications on Riemannian Manifolds with Ricci Curvature Bounds," Mathematics and Society, (2026): -, doi: 10.22108/msci.2026.146917.1763
VANCOUVER
Sohrabpour, M., Hajiaghasi, S., Azami, S. Comparison Theorems and Their Applications on Riemannian Manifolds with Ricci Curvature Bounds. Mathematics and Society, 2026; (): -. doi: 10.22108/msci.2026.146917.1763
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