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In this work, we study the rigidity problem for the logarithmic Sobolevinequality on a complete metric measure space $(M^n,g,f)$ with Bakry-\'EmeryRicci curvature satisfying $Ric_f\geq \frac{a}{2}g$, for some $a>0$. We provethat if equality holds then $M$ is isometric to $\Sigma\times \mathbb{R}$ forsome complete $(n-1)$-dimensional Riemannian manifold $\Sigma$ and by passingan isometry, $(M^n,g,f)$ must split off the Gaussian shrinking soliton$(\mathbb{R}, dt^2, \frac{a}{2}|.|^2)$. This was proved in 2019 by Ohta andTakatsu. In this paper, we prove this rigidity result using a different method.

Rigidity for the logarithmic Sobolev inequality on complete metric measure spaces