I have a valid US passport. I will fly Jet Airways to Delhi airport with 9 hours of layover to change to another Jet Airways Flight to Abu Dhabi. Do I need a transit visa?
If you're flying from the US via Delhi to Abu Dhabi with a US passport, you do NOT need a transit visa in India. Anything layover below 24 hours is covered under TWOV (transit without visa) rules.
Jet Airways staff is pretty helpful with regards to this: tell the ground staff on arrival in Delhi that you have a connecting flight and they will point you towards the transit area for international travellers.
You don't need any visa.
Just go to transit desk tell them what flight you arrived on and what flight you depart on and they will transfer your luggage for you. Once they (transfer desk staff) will contact someone from depart airlines, they will give you boarding pass (it can take a while), so you will be able to move on forward to duty free area and the gates.
If you don't leave the airport why would you need a transit visa?
I've had long layovers in various airports in the world and if you don't try to leave the airport there's no reason for someone to treat you like you're entering the country.
Sorry you're spending 9 hours in Delhi airport though... the city has lots of cool sites but the airport is NOT one of them. If I were you I might want to get a visa just to make decent use of that time rather than spending it in a sorry airport.