I will show that any Schubert or Richardson variety R in a flag manifold G/P is equivariantly rigid and convex. Equivariantly rigid means that R is uniquely determined by its equivariant cohomology class, and convex means that R contains any torus-stable subvariety whose fixed points belong to R. This is applied to prove that any two-pointed curve neighborhood representing a quantum cohomology product with a Seidel class is an explicitly determined Schubert variety, as well as Seidel multiplication formula in the equivariant quantum K-theory of any cominuscule flag variety. This is joint work with Pierre-Emmanuel Chaput and Nicolas Perrin.