が indicates the subject (except when used with some specific verbs/adjectives).
知る is a punctual verb (瞬間動詞), so ~ている form expresses the current state that results from prior movements (got knowledge about the city in the past, and as a result know it now).
知りますか is future tense.
In conclusion, マンチェスター市が知りますか? means "Will the city of Manchester know it?".
FYI, マンチェスターの市を知っていますか is usually interpreted as マンチェスターの市[いち], thus, this sentence means "Do you know a market in the city of Manchester?"