Complex differential geometry

What do distances we measure on a surface tell us about how it curves? As it turns out, quite a bit. And if we know how it curves everywhere, can we determine if it has a hole like a bagel or two or three holes like a pretzel? If we consider all the shapes a balloon can take, is the perfectly spherical one, the one whose curvature is most equally distributed everywhere? Can we tell if we are walking on a surface whether we can continue forever, or are bound to fall off it at the horizon? In differential geometry, we explore such questions on spaces that may have more than just the two dimensions that surfaces have. Complex geometry explores the same topics on spaces that are constrained in ways that are related to the invention of complex numbers.