Some browsers, like Firefox, can do it automatically. Most others don't.
I'd suggest one of three things:
1.) Have check_fields.php show the form again with the errors and the values of the $_POST array already filled in
2.) Use Javascript to check that each form item has a value, and if not throw an alert() and don't let them on to the next page.
3.) Combine your first form and check_fields.php so that it's one page and the form posts back to itself, then use the solution in #1.