Tangent space

Types of tangent space

Name Symbol Definition Tangent "Vector"
Geometric tangent space [ilmath]G_p(\mathbb{R}^n)[/ilmath][1] The set of tangents to a point in [ilmath]\mathbb{R}^n[/ilmath]

$G_p(\mathbb{R}^n)=\{(p,v)|v\in\mathbb{R}^n\}$ - the set of all arrows at [ilmath]p[/ilmath]

$v\in G_p(\mathbb{R}^n)\iff v=(u,p)\text{ for }u\in\mathbb{R}^n$ - pretty much just a vector
Tangent space (to [ilmath]\mathbb{R}^n[/ilmath]) [ilmath]T_p(\mathbb{R}^n)[/ilmath] The set of all derivations at [ilmath]p[/ilmath]]]

$\omega\in T_p(\mathbb{R}^n)\iff \omega:C^\infty(\mathbb{R}^n)\rightarrow\mathbb{R}$ is a derivation

Tangent vector
Tangent space (to a smooth manifold [ilmath]M[/ilmath]) [ilmath]T_p(M)[/ilmath] The set of all derivations at [ilmath]p[/ilmath], here a derivation is an [ilmath]\mathbb{R} [/ilmath]-linear map, $\omega:C^\infty(M)\rightarrow\mathbb{R}$ which satisfies the Leibniz rule. Recall [ilmath]C^\infty(M)[/ilmath] is the set of all smooth functions on our smooth manifold Tangent vector (to a manifold)
Tangent space (in terms of germs) [ilmath]\mathcal{D}_p(M)[/ilmath] The set of all derivations of [ilmath]C^\infty_p(M)[/ilmath] - the set of all germs of smooth functions at a point, that is:

$\omega\in \mathcal{D}_p(M)\iff\omega:C^\infty_p(M)\rightarrow\mathbb{R}$ is a derivation
See: Set of all derivations of a germ at a point

See

Geometric Tangent Space

The Geometric tangent space to [ilmath]\mathbb{R}^n[/ilmath] at [ilmath]p[/ilmath][2] is defined as follows:

• $G_p(\mathbb{R}^n)=\{(p,v)|v\in\mathbb{R}^n\}$ - the set of all arrows rooted at [ilmath]p[/ilmath]

Vector space

This is trivially a vector space with operations defined as follows:

• $v_p+w_p=(v+w)_p$
• $c(v_p)=(cv)_p$

Notations

• John M Lee uses [ilmath]\mathbb{R}^n_p[/ilmath] to mean the same thing ( [ilmath]G_p(\mathbb{R}^n)[/ilmath] )

Tangent Space

The Tangent space to [ilmath]\mathbb{R}^n[/ilmath] at [ilmath]p[/ilmath][3] is defined as follows:

• $T_p(\mathbb{R}^n)=\{\omega:\omega\text{ is a}$ derivation $\text{at }p\}$ - that is:
$\omega\in T_p(\mathbb{R}^n)\iff\omega:C^\infty(\mathbb{R}^n)\rightarrow\mathbb{R}$ where

Isomorphism between geometric tangent space and tangent space

Infact the geometric tangent space and tangent space to [ilmath]\mathbb{R}^n[/ilmath] at [ilmath]p[/ilmath] are linearly isomorphic to each other.

Proposition:

• $\alpha:G_p(\mathbb{R}^n)\rightarrow T_p(\mathbb{R}^n)$ given by:
• $\alpha:v_p\mapsto [D_v|_p:C^\infty(\mathbb{R}^n)\rightarrow\mathbb{R}]$
is a linear isomorphism

Theorem: The map $\alpha:G_p(\mathbb{R}^n)\rightarrow T_p(\mathbb{R}^n)$ given by $\alpha:v_p\mapsto [D_v|_p:C^\infty(\mathbb{R}^n)\rightarrow\mathbb{R}]$ is a linear isomorphism

TODO: ITSM p53 if help needed, uses LM has kernel of dim 0 [ilmath]\implies[/ilmath] injective

Tangent Space to a Manifold

The tangent space to a manifold [ilmath]M[/ilmath] at [ilmath]p[/ilmath] is defined as follows:

• $T_p(M)=\{\omega:\omega\text{ is a}$ derivation $\text{at }p\}$ - that is:
$\omega\in T_p(M)\iff\omega:C^\infty(M)\rightarrow\mathbb{R}$ where

Recall [ilmath]C^\infty(M)[/ilmath] is the set of all smooth functions on a smooth manifold [ilmath]M[/ilmath]

OLD PAGE

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[4] 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 $T_p(\mathbb{R}^n)=\left\{(p,v)|v\in\mathbb{R}^n\right\}$

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

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]
• $v|_a$

Why ordered pairs

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

Vector space

$T_p(A)$ 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 $\{e_1|_p,\cdots, e_n|_p\}$ and that the tangent space [ilmath]T_p(A)[/ilmath] is basically just a copy of [ilmath]A[/ilmath]