What is the ruling on lying to one's mother in order not to reveal to her one's worries and sorrows, and is expiation due after her mother said to her: "I beg you by God to tell me what is wrong with you?"

You are not liable for expiation nor sin, because what you uttered was not an oath, and what your mother said was a question, not an oath. Sheikh al-Islam Ibn Taymiyyah said: "When someone says, 'I ask you by Allah to do such-and-such,' this is a question and not an oath." You should have honored your mother's oath or answered her question. Asking by Allah is forbidden. If your mother intended to swear you to an action and you did not inform her, then she owes expiation for an oath.