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In this paper we discuss the local rigidity theorems with respect to scalarcurvature by using the Ricci flow. Let $M$ be an $n$-dimensional manifold and$V$ is an open subset in $M$. We prove that if $V$ satisfies that the scalarcurvature is non-negative and isoperimetric constant is no less than that ofEuclidean space, then $V$ is flat. We also obtain that if the Perelman's local$\nu$-entropy is non-negative for $V$ , then $V$ must be flat. As a corollory,we can improve a rigidity result obtained by Bakry, Concordet and Ledoux\cite{BCL} and Ni\cite{Ni2}, and show that if $V$ has the non-negative scalarcurvature and the $L^2$-logarithmic Sobolev inequality with the best constantof Euclidean space, then $V$ must be flat.

On local rigidity theorems with respect to the scalar curvature