The invention concerns a connection device (14) for connecting a turbomachine member such as an injector, to a system for supplying a fluid such as fuel, the connection device (14) comprising several concentric tubes (26, 28, 30) delimiting conduits (20, 22, 24) for supplying the member, which are curved in at least one direction, characterised in that it is made from a single part comprising at least the tubes (26, 28, 30).