In　this　paper,　we　examine　the　volume　comparison　theorem　associated　with　sigma　k\documentclass[12pt]{minimal}　\usepackage{amsmath}　\usepackage{wasysym}　\usepackage{amsfonts}　\usepackage{amssymb}　\usepackage{amsbsy}　\usepackage{mathrsfs}　\usepackage{upgreek}　\setlength{\oddsidemargin}{-69pt}　\begin{document}$$\sigma　_k$$\end{document}-curvature.　Specifically,　we　demonstrate　that　the　volume　comparison　theorem　with　respect　to　sigma　k\documentclass[12pt]{minimal}　\usepackage{amsmath}　\usepackage{wasysym}　\usepackage{amsfonts}　\usepackage{amssymb}　\usepackage{amsbsy}　\usepackage{mathrsfs}　\usepackage{upgreek}　\setlength{\oddsidemargin}{-69pt}　\begin{document}$$\sigma　_k$$\end{document}-curvature　is　valid　for　metrics　closed　to　strictly　stable,　positive　Einstein　metrics.　Utilizing　analogous　techniques,　we　derive　a　local　rigidity　theorem　for　strictly　stable　Ricci-flat　manifolds　with　respect　to　sigma　k\documentclass[12pt]{minimal}　\usepackage{amsmath}　\usepackage{wasysym}　\usepackage{amsfonts}　\usepackage{amssymb}　\usepackage{amsbsy}　\usepackage{mathrsfs}　\usepackage{upgreek}　\setlength{\oddsidemargin}{-69pt}　\begin{document}$$\sigma　_k$$\end{document}-curvature.　This　theorem　establishes　that　there　are　no　metrics　with　positive　sigma　k\documentclass[12pt]{minimal}　\usepackage{amsmath}　\usepackage{wasysym}　\usepackage{amsfonts}　\usepackage{amssymb}　\usepackage{amsbsy}　\usepackage{mathrsfs}　\usepackage{upgreek}　\setlength{\oddsidemargin}{-69pt}　\begin{document}$$\sigma　_k$$\end{document}-curvature　in　the　vicinity　of　strictly　stable　Ricci-flat　metrics.