Does anyone know if it is possible to restart Tomcat5.5 without actually killing the established connections?
I don't think so. What did you mean by connections? You meant database connections, as you can't quite hold persistent web connections (that exist for very long time)?
You may probably do so for hot-deployed web application. But this is only if the connection is maintained by Tomcat (as configured in the container) rather than initiated in a specific webapp. In that case redeploying the webapp should not affect the db connection.
For restarting Tomcat to effectuate a config change, for example, I don't think it can be done without stopping it at all.