Daf 10A says we can sleep in a bed with a covering if it has only 2 polls, but not if it has 4 because with 2 polls there is no flat surface a tefach wide. Is this the formal definition of a tent? If so, why the ban on umbrellas?
Yes, one of the requirements to be considered a tent with regard to the laws of Tum'ah is that the shelter have an area of at least one square Tefach. This is also the requirement with regard to the laws of Shabbos (see SHULCHAN ARUCH OC 315:11). For this reason, it is prohibited to open an umbrella on Shabbos, because one thereby creates a tent.
Although the surface of the umbrella is not flat but is slightly curved, we have learned (Shabbos 138a) that as long as there is a one Tefach by one Tefach space enclosed within three Tefachim of the top of the roof, it is still considered a tent. Thus, an umbrella -- which certainly has more than one square Tefach of space within three Tefachim of the top of the umbrella -- is considered a tent (Shulchan Aruch OC 315:8; see Bi'ur Halachah there, DH Tefach). Tosfos (Shabbos top of 138b) points out that although in Sukah 19b we seem to rule that Shipu'ei Ohalim are not like Ohalim, i.e. an Ohel must have distinquishable walls and roof, nevertheless he shows that if there is
a one Tefach by one Tefach space enclosed within three Tefachim of the top of the roof it is indeed considered an Ohel.
-Yisrael