Rigidity for the logarithmic Sobolev inequality on complete metric measure spaces

In this work, we study the rigidity problem for the logarithmic Sobolev inequality on a complete metric measure space (M-n,g,f)with Bakry-Emery Ricci curvature satisfying Ric(f) >= a/2g for some a>0.We prove that if equality holds, then Mis isometric to Sigma xR for some complete (n-1)-dimensional Riemannian manifold Sigma and by passing an isometry, (M-n,g,f) must split off the Gaussian shrinking soliton(R, dt(2), /vertical bar center dot vertical bar(2)). This was proved in 2019 by Ohta and Takatsu (ManuscrMath 162:271-282, 2019). In this paper, we prove this rigidity result using a different method.