Abstract :

From the elementary course in analysis, one knows the notion for continuity and differentiability for functions of real numbers. The notion of differentiability gets easily extended to functions $\mathbb{R}^n$ to $\mathbb{R}^m $, while the notion of continuity can be extended to arbitrary spaces with a certain structure, known as Topology. It is natural to ask: how can we generalize the notion of differentiability on arbitrary spaces? In this talk, we’ll discuss some, not arbitrary, topological spaces which are called smooth manifolds, that permits us to talk about the differentiability of functions. We’ll say how to define differentiable maps on a manifold. Since in most of the cases, manifolds won’t be linear spaces and we know derivatives are linear maps, one then asks how to define the derivative of a function at a point in such a setup. For that, we’ll discuss what is called a tangent space to a manifold at some point and push forward of a smooth map. We’ll try to conclude with vector bundles and push forward of vector fields if time permits.