  In this short note, we show the rigidity of a trace estimate for Steklov
eigenvalues with respect to functions in our previous work (Trace and inverse
trace of Steklov eigenvalues. J. Differential Equations 261 (2016), no. 3,
2026--2040.). Namely, we show that equality of the estimate holds if and only
if the manifold is a direct product of a round ball and a closed manifold. The
key ingredient in the proof is a decomposition theorem for flat and totally
geodesic Riemannian submersions which may be of independent interests.
