Introduction to Smooth Manifolds

Abstract

Manifolds crop up everywhere in mathematics. These generalizations of curves and surfaces to arbitrarily many dimensions provide the mathematical context for un- derstanding “space” in all of its manifestations. Today, the tools of manifold theory are indispensable in most major subfields of pure mathematics, and are becoming increasingly important in such diverse fields as genetics, robotics, econometrics, statistics, computer graphics, biomedical imaging, and, of course, the undisputed leader among consumers (and inspirers) of mathematics—theoretical physics. No longer the province of differential geometers alone, smooth manifold technology is now a basic skill that all mathematics students should acquire as early as possible.