Hi,
a riding arena (carriere equestre) is not a recreation ground, it should be tagged leisure=pitch + sport=equestrian, see https://wiki.openstreetmap.org/wiki/Tag:leisure%3Dpitch#Examples
and also
https://wiki.openstreetmap.org/wiki/Tag:building%3Dstable and
https://wiki.openstreetmap.org/wiki/Tag:building%3Driding_hall
