The Chairman entrust to you a mission as part of your work informally and asked you not to tell your hierarchical manager. What would you do ?

It depend on the trust level to the hierarchical manager. Fact is that every team or partnership depend about trust and this question is a question about our conscience. If the relation to the manager is very good I would say it will be good to talk with him, of course if the trust level is high enough, because maybe it could be a test!?