When we first install Tomcat, there is the default greeting page. I want to know if I should leave(make it stay as it is, or make no changes to) it as it is. Or should I do something about it? Thank you for your reply.
While you should never rely soley on 'security by obscurity', it certanly can't hurt to hide every detail possible about the system you're running.
Most well written sites will provide their own 404 and 500 error pages. Any default pages will have information about the site or company that the site was written for not the server that is being used to run the site.
So I would say Yes, get rid of that page in your production environment. Why hand a would be cracker the make and version of the container he is trying to break into?
While you're at it, get rid of any of the apps that ship with Tomcat if you're not using them (Balancer, documentation, examples, etc..). If you're going to use the manager app, it wouldn't hurt to rename it so someone can't just type "http://www.yourdomain.com/manager" to find out if you're using Tomcat.