Manège is the correct word. Menage is something else entirely.
Have had this discussion before and a lot of people say "oh it's just the English spelling". No, it isn't, it's not an English word and it doesn't have an English spelling - it's like a restaurant or a cul-de-sac or an abbatoir or a baguette... if you have spelt it differently you have spelt it WRONG :wink:
Sorry to add to the confusion :giggle: but here they actually use two different words - a manège is specifically an INDOOR arena, an outdoor arena is a carrière. But a manège is also the name for an entire riding establishment, and for the art of horsemanship itself.
So yeah, being English, why not just say "school" or "arena" and save everyone the trouble of correcting you :giggle: