How to stop the Mail app from running every 10 minutes on the Mac? I have Mountain Lion on an 27" iMac, and recently, for some reason, the Mail app launches every 10 minutes or so, and I have to quit it every time.

In the Preferences of Mail, I already tried every possible way to tell it not to start itself, but every 10 minutes or so, it still launches and I have to quit it manually.  I don't know whether it is because I set in Gmail calendar to send email alert on some special events in my Google Calendar, but that should be handled by the Gmail server, not the Mail app... 

I don't know whether it is related to the Notification Center, as the last few times, I notices messages popping up at the top right corner of the screen, and the Mail just started to run all by itself as well.  

Is there a way to stop it from running automatically?
This is not normal behavior. Edit your question with your console log information from the moment of launch, it might help.
Notification center will provide alerts when mail arrives if set to do so, and clicking on a message will launch Mail. Mail messages also reside in the notification center and clicking one there will also launch mail.
Short of these events, Mail should not launch itself, unless you have some background process or script you are unaware of.

There is a background process or helper to allow the notification center to show incoming messages while Mail.app itself isn't running. This is normal behavior. If Mail is turned off in the notification system preference you won't have any intrusions.

Of course different things such as shell scripts, other apps and so on can launch apps so if you've configured other apps to launch Mail you need to deal with that scenario. This is where the log files can give you a hint as to what happened immediately prior to Mail launching.