Rigidity is a quality found in people and objects that don't bend — though they might eventually break.
When we see rigidity in a person, it means they're severe, like a teacher who punishes you for being late even though you were busy saving an orphan from a polar bear. Rigidity means no bending — in that case, of the rules. Objects and substances can have rigidity as well, if they literally won't bend. Rigidity means about the same thing as inflexibility, another word that describes rock-solid, unyielding people and substances.