  We investigate the rigidity problem for the logarithmic Sobolev inequality on
weighted Riemannian manifolds satisfying $\mathrm{Ric}_{\infty} \ge K>0$.
Assuming equality holds, we show that the $1$-dimensional Gaussian space is
necessarily split off, similarly to the rigidity results of Cheng--Zhou on the
spectral gap as well as Morgan on the isoperimetric inequality. The key
ingredient of the proof is the needle decomposition method introduced on
Riemannian manifolds by Klartag. We also present several related open problems.
