Understanding Tangent Spaces

In this post, we’ll explore one of the most fundamental concepts in differential geometry: tangent spaces. We’ll see how these objects naturally arise from our desire to do calculus on manifolds, and how they generalize our intuition from calculus in ℝⁿ.

What is a Tangent Space?

At each point p of a smooth manifold M, we can define a vector space TpM called the tangent space. This space contains all possible “directions” in which we can move from p while staying on the manifold. Think of it as the best linear approximation to M at p.

Key Concepts We’ll Cover:

1. Definition of tangent vectors as derivations
2. The relationship between different viewpoints on tangent vectors
3. How to compute tangent spaces in practice
4. Examples in ℝⁿ and on spheres

Stay tuned as we develop these ideas further in upcoming posts!