The phrase "in an arch manner" can be replaced with several synonyms to convey a similar meaning. Some of the possible options include "sardonically," "mockingly," "ironically," "satirically," "tauntingly," "sarcastically," "facetiously," or "whimsically." All of these adverbs describe a kind of behavior that is marked by a certain level of insincerity or amusement, often accompanied by a sense of superiority or irony. They can be used to describe the way someone speaks, gestures, or expresses themselves, especially when they are trying to be clever or witty in a somewhat provocative or teasing manner.