Both my wrecked and my replacement Metro (2001 LSi, 4D, auto) have this small hose mounted on a clamp on the upper radiator hose with a cap on one end. The old one was hooked up to something, the new Metro doesn't have it hooked up to anything. I can't look at the wrecked one any more, the crummy manual I have doesn't talk about it, and I can't recall what it is and what it's supposed to be hooked up to, though I recall it had to do with the tranny, maybe some kind of vent or overflow. Maybe some part is missing? Someone else with a Metro mind having a look and tell me where that hose is supposed to be attached?
it looks like some sort of a vent. Does the other end go to the engine block? Could the oily residue have leaked from that little cap? Maybe its related to the PCV system. Just a guess... Maybe its the transaxle vent.
Yes it is a transaxle vent. I can not see in your pic, but there is a small brass fitting near the top of the tranny that the hose plugs into. The fitting may be near the fluid fill pipe on your automatic.
Hope that helps.