Tangent Space

The vector space of tangent vectors at a point, defined intrinsically using derivations or curves.

Tangent Space

Let $M$ be a smooth manifold and $p\in M$ . The tangent space at $p$ , denoted $T_pM$ , is a vector space capturing first-order directions through $p$ .

Derivation definition

Let $C^\infty_p(M)$ denote the germs of smooth real-valued functions near $p$ . A tangent vector at $p$ is a linear map

v: C^\infty_p(M)\to \mathbb{R}

such that $v$ satisfies the Leibniz rule

v(fg)=v(f)\,g(p)+f(p)\,v(g).

The set of all such derivations is $T_pM$ , with addition and scalar multiplication defined pointwise.

Coordinate description

Given a chart $(U,x)$ with $p\in U$ and $x=(x^1,\dots,x^n)$ , there are distinguished tangent vectors $\left.\frac{\partial}{\partial x^i}\right|_p$ defined by

\left.\frac{\partial}{\partial x^i}\right|_p(f)=\frac{\partial (f\circ x^{-1})}{\partial u^i}\Big|_{u=x(p)}.

These form a basis of $T_pM$ , so $\dim T_pM = \dim M$ .

Curve viewpoint

Equivalently, $T_pM$ can be described using equivalence classes of smooth curves $\gamma:(-\epsilon,\epsilon)\to M$ with $\gamma(0)=p$ , where $\gamma_1\sim\gamma_2$ if they have the same first derivative in any (hence every) chart.

Example

If $M=\mathbb{R}^n$ , then $T_p\mathbb{R}^n\cong \mathbb{R}^n$ canonically, and derivations correspond to directional derivatives (see derivative ).