How does Claudius manipulate Gertrude?

Claudius manipulates Gertrude, his wife, by not being completely (or even minimally) honest with her. For obvious reasons, he does not tell her that he murdered his brother, her first husband, and Prince Hamlet's father. He somehow gets her to marry him really quickly after his brother's death, and one might imagine some manipulation to be involved in this circumstance as well because she does not wait out the proper period of mourning before remarrying. Further, Claudius isn't completely honest with his wife concerning why he brings Rosencrantz and Guildenstern to Denmark; he says that he only wants to find out the cause of Hamlet's upset so that he can help to remedy it, but he seems to actually be a great deal more suspicious of Hamlet than that based on the fact that he speaks to Hamlet so formally, insults his "unmanly" grief, and will not let him return to Wittenberg.
Moreover, on the night of the play that so upsets Claudius, he approves Polonius's idea to secretly listen in on the conversation Hamlet has with Gertrude, putting Gertrude into a pretty uncomfortable position. She obviously doesn't want her son to know that Polonius is spying on them, but she also seems to feel that she is somewhat mistrusted, or else why would there be a need for someone to monitor their conversation, as she tells Polonius, "Fear me not," or, in other words, "don't doubt me" (3.4.9).