A domain name is a dot-separated list of strings containing letters, digits and minus signs, like pathsurvey.eu.org. An URL is something like http://www.eu.org/rrtypes.html. Only the part between http:// and the next / is a domain name.
EU.org only deals with domain names, not URLs. In short, this means that if you only want to give your Web homepage a shorter or nicer name, DNS in EU.org is not for you.