Tangent space

Types of tangent space

Name	Symbol	Definition	Tangent "Vector"
Geometric tangent space	[ilmath]G_p(\mathbb{R}^n)[/ilmath]	The set of tangents to a point in [ilmath]\mathbb{R}^n[/ilmath] [math]G_p(\mathbb{R}^n)=\{(p,v)|v\in\mathbb{R}^n\}[/math] - the set of all arrows at [ilmath]p[/ilmath]	[math]v\in G_p(\mathbb{R}^n)\iff v=(u,p)\text{ for }u\in\mathbb{R}^n[/math] - 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]]] [math]\omega\in T_p(\mathbb{R}^n)\iff \omega:C^\infty(\mathbb{R}^n)\rightarrow\mathbb{R} [/math] 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, [math]\omega:C^\infty(M)\rightarrow\mathbb{R}[/math] 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: [math]\omega\in \mathcal{D}_p(M)\iff\omega:C^\infty_p(M)\rightarrow\mathbb{R}[/math] is a derivation See: Set of all derivations of a germ at a point

See

• Motivation for tangent space definitions
• Motivation for tangent space

Geometric Tangent Space

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

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

Vector space

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

• [math]v_p+w_p=(v+w)_p[/math]
• [math]c(v_p)=(cv)_p[/math]

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] is defined as follows:

• [math]T_p(\mathbb{R}^n)=\{\omega:\omega\text{ is a}[/math] derivation [math]\text{at }p\}[/math] - that is:

  [math]\omega\in T_p(\mathbb{R}^n)\iff\omega:C^\infty(\mathbb{R}^n)\rightarrow\mathbb{R}[/math] where

  • [ilmath]\omega[/ilmath] is [ilmath]\mathbb{R} [/ilmath]-linear
  • [ilmath]\omega[/ilmath] satisfies the Leibniz rule

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:

• [math]\alpha:G_p(\mathbb{R}^n)\rightarrow T_p(\mathbb{R}^n)[/math] given by:
  • [math]\alpha:v_p\mapsto [D_v|_p:C^\infty(\mathbb{R}^n)\rightarrow\mathbb{R}][/math]

is a linear isomorphism

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

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

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Tangent Space to a Manifold

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

• [math]T_p(M)=\{\omega:\omega\text{ is a}[/math] derivation [math]\text{at }p\}[/math] - that is:

  [math]\omega\in T_p(M)\iff\omega:C^\infty(M)\rightarrow\mathbb{R}[/math] where

  • [ilmath]\omega[/ilmath] is [ilmath]\mathbb{R} [/ilmath]-linear
  • [ilmath]\omega[/ilmath] satisfies the Leibniz rule

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

See also

• Differential of a smooth map - the differential of a smooth map - a map between tangent spaces of manifolds

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 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

See also Motivation for 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