Differentiable Manifolds

A very, very broad class of problems in modern mathematics can be phrased as: if we know local information about an object, what global statements can we make about it? For example, the study of differential equations aims to answer if we know how a function behaves locally (since that is exactly what the derivative tells us), can we find what the function is exactly? Or more generally what properties of the solution can be found by considering the differential equation itself? In a similar vein, one can think about the study of differentiable manifolds. In this case, we are given a topological space and we know exactly what this space looks like on small patches and we even know how these patches stitch together. Then, broadly speaking, the general motivating question is what global statements can we make about our space.

Two coordinate charts on a manifold.

Differentiable Manifolds

Table of Contents

• The definition of differentiable manifolds
• Functions on differentiable manifolds
• Functions of differentiable manifolds
• Maximal atlases
• Vector bundles
• The tangent and cotangent spaces
• Tensor fields
• Lie groups
• Differential forms
• Vector fields along curves
• De Rham cohomology
• The Mayer–Vietoris sequence of De Rham cohomology
• Orientation
• Integration of forms