I own a property (rental property)my mother manages and collects all the rent and she keep it.Who is in charge of reporting that as earned income?

If you own the property then it is your income, you would report it.

If you let her keep it, then you are gifting it to her or paying her for the work she does.

You need to issue her a 1099 for the income she gets if it for managing the property and she would pay taxes on it.

You would then deduct what you pay her on the Rental schedule on your return.