To use the Calendar app on iOS you need to configure your account.
On your iPhone or iPad go to the Settings app.
Scroll down and pick Calendar. Then Choose accounts.
Then Add and Account.
Pick ‘Add CalDAV Account’
Fill out the form as follows:
User Name: <email@example.com> (It’s the same as you log into zimbra with).
Password: Your DMS/Zimbra password.
Description: Since this links your calendar account call it something like ‘work calendar’. Your shared calendars will get their proper name.
The synchronizing of the events will take a few minutes.
New appointments or changes to appointments will take a few minutes to replicate to other devices. Be patient.