  We investigate the rigidity problem for the sharp spectral gap on Finsler
manifolds of weighted Ricci curvature bound $\text{Ric}_{\infty} \geq K > 0$.
Our main results show that if the equality holds, the manifold necessarily
admits a diffeomorphic splitting (or isometric splitting in the particular
class of Berwald spaces). This splitting phenomenon is comparable to the
Cheeger-Gromoll type splitting theorem by Ohta. We also obtain the rigidity
results of logarithmic Sobolev and Bakry-Ledoux isoperimetric inequalities via
needle decomposition as corollaries.
