I think you would use "this is Anna" specifically if you have communicated with your professor previously but not by e-mail (in person, by phone, etc.), and this professor may not recognize your identity from your e-mail address alone.
If you were introducing yourself to a new person, you might say something like "I am Anna," but I'd expect that people would be more used to "Greetings, My name is Anna."
Keep in mind that anyone who receives e-mail from you already knows who you are. E-mail has a "from" field; with most e-mail software, it displays your real name as well as your e-mail address. Also, you almost certainly will sign your name at the end. Unless the e-mail is very long, anyone who reads it will see your name there. That makes such an opening unnecessary. It is seldom used in business e-mail, unless one wants to say something that is not obvious from the other sources of this information. When my students open an e-mail to me that way, it usually means that they are trying to be polite but don't know how.