Tangent space

I prefer to denote the tangent space (of a set [ilmath]A[/ilmath] at a point [ilmath]p[/ilmath]) by [ilmath]T_p(A)[/ilmath] - as this involves the letter T for tangent however one author uses [ilmath]T_p(A)[/ilmath] as Set of all derivations at a point - the two are indeed isomorphic but as readers will know - I do not see this as an excuse.

What is defined here may also be called the Geometric tangent space

Definition

It is the set of arrows at a point, the set of all directions essentially. As the reader knows, a vector is usually just a direction, we keep track of tangent vectors and know them to be "tangent vectors at t" or something similar. A tangent vector is actually a point with an associated direction.

Euclidean (motivating) definition

We define [math]T_p(\mathbb{R}^n)=\left\{(p,v)|v\in\mathbb{R}^n\right\}[/math]

Generally then we may say: [math]T_p(A)=\left\{(p,v)|v\in A\right\}[/math]

Notation

A tangent vector (often [ilmath]v[/ilmath] is used) shall be left as just [ilmath]v[/ilmath] if the point to which it is a tangent to is implicit (ie "[ilmath]v[/ilmath] is a tangent at [ilmath]p[/ilmath]")

Rather than writing [ilmath](p,v)[/ilmath] we may write:

• [ilmath]v[/ilmath] (if it is implicitly understood that this is a tangent to the point [ilmath]p[/ilmath])
• [ilmath]v_a[/ilmath]
• [math]v|_a[/math]

Why ordered pairs

Ordered pairs are used because now the tangent space at two distinct points are disjoint sets, that is [math]\alpha\ne\beta\implies T_\alpha(A)\cap T_\beta(A)=\emptyset[/math]

Vector space

[math]T_p(A)[/math] is a vector space when equipped with the following definitions:

• [ilmath]v_a+w_a=(v+w)_a[/ilmath]
• [ilmath]c(v_a)=(cv)_a[/ilmath]

It is easily seen that the basis for this is the standard basis [math]\{e_1|_p,\cdots, e_n|_p\}[/math] and that the tangent space [ilmath]T_p(A)[/ilmath] is basically just a copy of [ilmath]A[/ilmath]

See also

• Set of all derivations at a point
• Set of all derivations of a germ
• The tangent space and derivations at a point are isomorphic