Unfortunately the film is without sound but I think Mike and SpitfireVb hit the nail on the head. A far distance marriage. "Only" the groom is getting the ring (none of the other two soldiers got a ring - as they are groomesmen). Therefore the groom is signing the mariage certificate as does the officer as marriage registrar. And as Spitfire Vb stated, here are wedding rings worn on the right hand.
Regards,