University of Isfahan Mathematics and Society 2345-6493 Articles in Press 2026 02 07 Comparison Theorems and Their Applications on Riemannian Manifolds with Ricci Curvature Bounds Comparison Theorems and Their Applications on Riemannian Manifolds with Ricci Curvature Bounds 30273 10.22108/msci.2026.146917.1763 FA Mahin Sohrabpour Department of Pure Mathematics, Faculty of Science, Imam Khomeini International University, Qazvin, Iran. Sakineh Hajiaghasi Department of Pure Mathematics, Faculty of Science, Imam Khomeini International University, Qazvin, Iran. Shahroud Azami Department of Pure Mathematics, Faculty of Science, Imam Khomeini International University, Qazvin, Iran. Journal Article 2025 10 09 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<br /><br />‎\begin{align}\nonumber‎<br /><br />‎\tilde{Ric}:= Ric‎ + ‎\frac{1}{2}\mathcal{L}_{V}g \geq (n-1)k,<br /><br />‎\end{align}‎<br /><br /><br /><br />and then for manifolds satisfying the Bakry–Émery Ricci curvature condition as follow<br /><br /><br /><br />‎\begin{equation}‎\nonumber‎<br /><br />‎Ric+Hess h\geq (n-1)k‎,<br /><br />‎‏‎\end{equation}<br /><br /><br /><br />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. 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<br /><br />‎\begin{align}\nonumber‎<br /><br />‎\tilde{Ric}:= Ric‎ + ‎\frac{1}{2}\mathcal{L}_{V}g \geq (n-1)k,<br /><br />‎\end{align}‎<br /><br /><br /><br />and then for manifolds satisfying the Bakry–Émery Ricci curvature condition as follow<br /><br /><br /><br />‎\begin{equation}‎\nonumber‎<br /><br />‎Ric+Hess h\geq (n-1)k‎,<br /><br />‎‏‎\end{equation}<br /><br /><br /><br />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. https://math-sci.ui.ac.ir/article_30273_77f5a9d3ebd5ee53fe8ffd971942bddc.pdf