Isometries are important in lots of areas of science. One such area is cosmology - the study of the universe. Cosmologists think of space-time itself - that is, the entire universe, which includes everything that ever exists and everything that ever will exist - as a kind of four-dimensional version of an elastic sheet.
Nobody knows whether this sheet is flat, curved in on itself like a sphere, or curved away from itself like a potato chip. The answer to this question will determine whether the universe will go on forever or whether it will end at some point in the distant future.
The shape of the universe is probably one of the most interesting open questions in science, and it involves spooky-sounding stuff such as dark matter and dark energy. Read here about human attempts to measure the shape of the universe: http://www.mathaware.org/mam/05/shape.of.universe.html
As well being important for very big things like the universe, the math of curved surfaces also works on very small scales. For example, a 2D crystal lattice, when grown on a curved surface, produces interesting and beautiful patterns that wouldn’t be possible see if the crystal were grown flat. http://news.uchicago.edu/article/2010/12/23/physicists-grow-pleats-two-dimensional-curved-spaces
Check out this shout out from Vsauce: https://www.youtube.com/playlist?list=PLZRRxQcaEjA5fgfW3a3Q0rzm6NgbmICtg