Tangent Space

Definition

For \(x \in \mathcal{M}\), the tangent space \(T_x\mathcal{M}\) of a manifold \(\mathcal{M}\) at point \(x\) is the first order linear approximation of \(\mathcal{M}\) around \(x\).

A tangent space is the “flat” space that approximates the manifold at a specific point, where we can do all the familiar operations of linear algebra, like addition and scalar multiplication.