While the patent was published today in the database of the USPTO (United States Patent and Trademark Office), the request was put in as early as September of last year.
There is little explanatory text, but it's clear the touchscreen display would run around the back of the smartphone - on which app icons could be displayed.
It's unclear whether we will see this appear in Samsung's next batch of smartphones, but with the industry-wide surge of bezel-less designs it's clear that the Korean company will need something new for the future of its phone designs.
And, of course, this may not even reach the smartphone stage. Samsung has also been rumoured to be making a bendable phone that could launch in 2018, so it's possible this will be the manufacturer's priority.
Still, if a curved phone means stronger glass (as it would be difficult to put a case around it) and, therefore, crack-proof screens, we're all for it.