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 understanding “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. Over the past century or two,mathematicians have developed a wondrous collection of conceptual machines that enable us to peer ever more deeply into the invisible world of geometry in higher dimensions. Once their operation is mastered, these powerful machines enable us to think geometrically about the 6-dimensional solution set of a polynomial equation in four complex variables, or the 10-dimensional manifold of 5 5 orthogonal matrices, as easily as we think about the familiar 2-dimensional sphere in R3. The price we pay for this power, however, is that the machinesareassembledfromlayeruponlayerofabstractstructure.Startingwiththe familiar raw materials of Euclidean spaces, linear algebra, multivariable calculus, and differential equations,one must progress through topological spaces,smooth atlases, tangent bundles, immersed and embedded submanifolds, vector fields, flows, cotangent bundles, tensors, Riemannian metrics, differential forms, foliations, Lie derivatives, Lie groups, Lie algebras, and more—just to get to the point where one can even think about studying specialized applications of manifold theory such as comparison theory, gauge theory, symplectic topology, or Ricci flow.