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Let $G=PSO(2n+1, \mathbb{C}) (n \ge 3)$ and $B$ be the Borel subgroup of $G$containing maximal torus $T$ of $G.$ Let $w$ be an element of Weyl group $W$and $X(w)$ be the Schubert variety in the flag variety $G/B$ corresponding to$w.$ Let $Z(w, \underline{i})$ be the Bott-Samelson-Demazure-Hansen variety(the desingularization of $X(w)$) corresponding to a reduced expression$\underline{i}$ of $w.$ In this article, we study the cohomology modules of thetangent bundle on $Z(w_{0}, \underline{i}),$ where $w_{0}$ is the longestelement of the Weyl group $W.$ We describe all the reduced expressions of$w_{0}$ in terms of a Coxeter element such that all the higher cohomologymodules of the tangent bundle on $Z(w_{0}, \underline{i})$ vanish (see Theorem\ref{theorem 8.1}).

Rigidity of Bott-Samelson-Demazure-Hansen variety for $PSO(2n+1, \mathbb{C})$