I’m interested in developing an iPhone application using Visual Studio and C#.NET on Windows Vista.
I tried to download the SDK from http://developer.apple.com/iphone
yes. monotouch works fine
Current answer to this question is yes, with MonoTouch from Novell using MonoDevelop on a Mac computer, but bear in mind that Novell laid off the Mono/MonoTouch guys, who are right now in the middle of setting up a startup to replace MonoTouch with a new product called Xamarin, due for release this summer.
January 2013
Xamarin came through with MonoTouch and Mono for Android quite a while ago now, both of which are great products, if somewhat expensive for the enthusiast.
or you could just run mac os on your pc http://oreilly.com/windows/archive/PearPC.html check it out
UPDATE 7/18/2013:
Since this answer is 3 years old I would like to update it with current information. MonoTouch is no more because it is now part of Xamarin. With it you can develop mobile applications using C# and parts of the .NET Framework (using Mono). You can develop iOS apps using Windows and Visual Studio, but you still need a Mac to compile and build. Here's a blog post on how to use VS for iOS development.
ORIGINAL ANSWER
MonoTouch. You still need a mac and a minimum of $400 (as of this writing)
You can try the eval version which doesn't require you to pay and allows you program using the simulator. If you want to test your app on the hardware then you need to pocket the money.
It is possible to develop for the iPhone in C# using Unity. It performs ahead of time compilation as described here. However, Unity seems more aimed at developing games and the iPhone SDK currently requires Mac OSX.
In a word, no. There is Mono support for the iPhone, but it's currently limited. They have spoken in the past about the possibility of writing a Silverlight app, with a series of "iPhone style" controls that could be statically linked as an iPhone app, but that's some way off.
If it's games you're interested in, and you have deep pockets, then UNITY is a Mono based game development platform that supports the iPhone (and runs on Windows)
Failing that, bite the bullet and get a cheap Mac from EBay. Officially only Intel Macs are supported for the SDK, but you can get it to install on PPC ones and I've yet to have a problem with that (I'm running it on a G5 PowerMac at the moment)