If you want to use RequestLocationUpdates method in LocationManager, you need to add a class which implements ILocationListener to receive the callback from LocationManager. Most of the examples on the internet used the Activity as ILocationListener. That is easy if you choose the implementation in this way. But I wish to build a self-contained class which does not inherit any UI class.
locationManager.RequestLocationUpdates(LocationProviders.GPS, 10, 10, this);
Firstly, I just implemented all methods from the interface. Then I got an invalid casting exception. It is because LocationManager required the class is a sub-class of Java.lang.Object. In the native Android SDK, all classes are sub-class of Java.lang.Object, but we are using Xamarin, that is .Net, not Java. Thus we need to inherit our class to be a Java Object. After I make my class to be a sub-class of Java.lang.The object, I got an exception about the invalid handle. That is because ILocationListener has a property
public IntPtr Handle

I built a property to get from Application Context
Application.Context.Handle

Actually, Handle Property is in the Java.lang.Object. I do not need to build my own, I need to remove the section. Then it works.