No, because kraeusen will stick to it and mess with the reading. Plus it's difficult to get it out without breaking it if you just let it free-float in a carboy.
I may just be OCD, but I don't want anything left in my beer but the yeast doing its work.
Lots of people "thief" samples for SG readings - I just sanitize and float my hydrometer in my fermenter (plastic bukets are very nice for this - obviously carboys are a different story).
My advice is to NOT leave the hydro in your fermenter.
this may be common knowledge, but I have discovered that the outer cylinder of your auto siphon can be used as a thief sampler.
remove the cane portion use the outer cylinder only. sanatize, then place bottom of tube under the surface of the beer. in a quick motion work the tube straight up and down. on the down stroke beer will come into the tube, and on the way up the check will close trapping the beer inside
Spend the 12 bucks or so on the fermtech thief. It fits a hydrometer in it and fits through all my carboys. One if the best things I've bought. Also great for getting samples. It is essentially a large bottling wand.
I think I may try the fishing line thing because I hate wasting beer and don't feel like drinking flat beer all the time either. But I don't think I am going to leave it in the fermenter though.