Real and complex manifolds lie at the core of the study of modern geometry. A real manifold (or smooth manifold) is a topological space which is locally homeomorphic to real n-dimensional space and whose transition functions are differentiable. A complex manifold is locally like an open set in complex n-dimensional space with holomorphic transition functions.
In many cases, a given topological space will posess only one smooth structure (of interest); however, the smooth structure allows us to bring multivariate calculus to bear on problems concerning the topology of the manifold.
A complex manifold, by contrast, often possesses a much richer structure than its underlying smooth manifold. For example, the set of all complex structures on a given real manifold varies continuously and can itself be given the structure of a complex manifold. Using a theorem of Kodaira, one can often embed compact complex manifolds algebraically in complex projective space, allowing for fruitful interaction with algebraic geometry.
In this course we will study the basic properties of real and complex manifolds. As examples of real manifolds, we will focus on Lie groups. Our basic examples of complex manifolds will be Riemann surfaces, also known as complex curves. We will also examine links with algebraic topology.
Prerequisites for the course are a knowledge of multivariate calculus, linear algebra, and basic complex analysis. Some familiarity with algebraic topology will be helpful, but not required.