ABSTRACT
curvature—seealsoGaussian curvature Every surface has an Euler number Euler number , an integer that contains essential information about the surface’s global topology. (“Euler” is pronounced “oiler”.) The Euler number is easy to compute, and it immediately predicts which homogeneous geometry the surface will admit: surfaces with positive Euler number admit spherical geometry, surfaces with zero Euler number admit flat geometry, and surfaces with negative Euler number admit hyperbolic geometry. In fact, the Euler number is so powerful that if you know a surface’s Euler number and you know whether it’s orientable or not, then you can immediately say what global topology the surface has! The Gauss-Bonnet formula Gauss-Bonnet formula relates a surface’s Euler number to its area and curvature. (“Bonnet” is a French name, so the ‘t’ is silent and the stress is on the second syllable, “buh-NAY”.)
