example.appspot.com does not transfer cookies from example.com - you will not be able to identify the user without logging in to example.appspot.com as well.
You could, of course, completely disable Google Authentication at example.appspot.com and implement your own scheme; you can add the signature and username to the AJAX requests that you create and verify the signature in the application application. if the signature is valid, just accept the user who was transferred as an authenticated user and pretended to be logged in.
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