Difference between revisions of "Tangent space"

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==Types of tangent space==
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{| class="wikitable" border="1"
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|-
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! Name
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! Symbol
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! Definition
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! Tangent "Vector"
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|-
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| [[Tangent space#Geometric Tangent Space|Geometric tangent space]]
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| {{M|G_p(\mathbb{R}^n)}}<ref>Alec's notation - John M Lee uses {{M|\mathbb{R}^n_p}} and it is distinct from {{M|T_p(\mathbb{R}^n)}}</ref>
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| The set of tangents to a point in {{M|\mathbb{R}^n}}<br/>
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<math>G_p(\mathbb{R}^n)=\{(p,v)|v\in\mathbb{R}^n\}</math> - the set of all arrows at {{M|p}}
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| <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
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|-
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| [[Tangent space#Tangent Space|Tangent space (to {{M|\mathbb{R}^n}})]]
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| {{M|T_p(\mathbb{R}^n)}}
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| The set of all [[Derivation|derivations]] at {{M|p}}]]<br/>
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<math>\omega\in T_p(\mathbb{R}^n)\iff \omega:C^\infty(\mathbb{R}^n)\rightarrow\mathbb{R} </math> is a [[Derivation|derivation]]
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| [[Tangent vector]]
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|-
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| [[Tangent space#Tangent Space to a Manifold|Tangent space (to a smooth manifold {{M|M}})]]
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| {{M|T_p(M)}}
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| The set of all [[Derivation|derivations]] at {{M|p}}, here a derivation is an [[Linear map|{{M|\mathbb{R} }}-linear map]], <math>\omega:C^\infty(M)\rightarrow\mathbb{R}</math> which satisfies the [[Leibniz rule]]. Recall {{M|C^\infty(M)}} is the set of all [[Smooth function|smooth functions]] on our [[Smooth manifold|smooth manifold]]
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| [[Tangent vector#On a Manifold|Tangent vector (to a manifold)]]
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|-
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| Tangent space (in terms of [[Germ|germs]])
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| {{M|\mathcal{D}_p(M)}}
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| The set of all [[Derivation|derivations]] of [[The set of all germs of smooth functions at a point|{{M|C^\infty_p(M)}} - the set of all germs of smooth functions at a point]], that is:<br/>
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<math>\omega\in \mathcal{D}_p(M)\iff\omega:C^\infty_p(M)\rightarrow\mathbb{R}</math> is a derivation<br/>
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'''See: ''' [[Set of all derivations of a germ at a point]]
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|}
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See
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* [[Motivation for tangent space definitions]]
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* [[Motivation for tangent space]]
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==Geometric Tangent Space==
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The '''Geometric tangent space to {{M|\mathbb{R}^n}} at {{M|p}}'''<ref>Introduction to smooth manifolds - John M Lee - Second Edition</ref> is defined as follows:
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* <math>G_p(\mathbb{R}^n)=\{(p,v)|v\in\mathbb{R}^n\}</math> - the set of all arrows rooted at {{M|p}}
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===Vector space===
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This is trivially a [[Vector space|vector space]] with operations defined as follows:
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* <math>v_p+w_p=(v+w)_p</math>
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* <math>c(v_p)=(cv)_p</math>
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===Notations===
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* John M Lee uses {{M|\mathbb{R}^n_p}} to mean the same thing ( {{M|G_p(\mathbb{R}^n)}} )
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==Tangent Space==
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The '''Tangent space to {{M|\mathbb{R}^n}} at {{M|p}}'''<ref>Introduction to smooth manifolds - John M Lee - Second Edition</ref> is defined as follows:
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* <math>T_p(\mathbb{R}^n)=\{\omega:\omega\text{ is a}</math> [[Derivation|derivation]] <math>\text{at }p\}</math> - that is:
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*: <math>\omega\in T_p(\mathbb{R}^n)\iff\omega:C^\infty(\mathbb{R}^n)\rightarrow\mathbb{R}</math> where
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*:* {{M|\omega}} is [[Linear map|{{M|\mathbb{R} }}-linear]]
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*:* {{M|\omega}} satisfies the [[Leibniz rule]]
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==Isomorphism between geometric tangent space and tangent space==
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Infact the geometric tangent space and tangent space to {{M|\mathbb{R}^n}} at {{M|p}} are [[Linear isometry|linearly isomorphic]] to each other.
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'''Proposition: '''
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* <math>\alpha:G_p(\mathbb{R}^n)\rightarrow T_p(\mathbb{R}^n)</math> given by:
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** <math>\alpha:v_p\mapsto [D_v|_p:C^\infty(\mathbb{R}^n)\rightarrow\mathbb{R}]</math>
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: is a [[Linear isometry|linear isomorphism]]
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{{Begin Theorem}}
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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 isometry|linear isomorphism]]
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{{Begin Proof}}
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{{Todo|ITSM p53 if help needed, uses LM has kernel of dim 0 {{M|\implies}} injective}}
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{{End Proof}}
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{{End Theorem}}
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==Tangent Space to a Manifold==
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The '''tangent space to a manifold {{M|M}} at {{M|p}}''' is defined as follows:
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* <math>T_p(M)=\{\omega:\omega\text{ is a}</math> [[Derivation|derivation]] <math>\text{at }p\}</math> - that is:
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*: <math>\omega\in T_p(M)\iff\omega:C^\infty(M)\rightarrow\mathbb{R}</math> where
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*:* {{M|\omega}} is [[Linear map|{{M|\mathbb{R} }}-linear]]
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*:* {{M|\omega}} satisfies the [[Leibniz rule]]
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Recall {{M|C^\infty(M)}} is the set of all [[Smooth function|smooth functions]] on a [[Smooth manifold|smooth manifold {{M|M}}]]
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==See also==
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* [[Differential of a smooth map]] - the differential of a [[Smooth map|smooth map]] - a map between tangent spaces of manifolds
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==OLD PAGE==
 
I prefer to denote the tangent space (of a set {{M|A}} at a point {{M|p}}) by {{M|T_p(A)}} - as this involves the letter T for tangent however one author<ref>John M. Lee - Introduction to Smooth Manifolds - second edition</ref> uses {{M|T_p(A)}} 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.
 
I prefer to denote the tangent space (of a set {{M|A}} at a point {{M|p}}) by {{M|T_p(A)}} - as this involves the letter T for tangent however one author<ref>John M. Lee - Introduction to Smooth Manifolds - second edition</ref> uses {{M|T_p(A)}} 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.
  
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*[[Set of all derivations at a point]]
 
*[[Set of all derivations at a point]]
 
*[[Set of all derivations of a germ]]
 
*[[Set of all derivations of a germ]]
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*[[The tangent space and derivations at a point are isomorphic]]
 
  
 
==References==
 
==References==

Latest revision as of 19:27, 14 April 2015

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]

[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

Geometric Tangent Space

The Geometric tangent space to [ilmath]\mathbb{R}^n[/ilmath] at [ilmath]p[/ilmath][2] 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][3] 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

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


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:

  • [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

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


See also

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

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

References

  1. ↑ Alec's notation - John M Lee uses [ilmath]\mathbb{R}^n_p[/ilmath] and it is distinct from [ilmath]T_p(\mathbb{R}^n)[/ilmath]
  2. ↑ Introduction to smooth manifolds - John M Lee - Second Edition
  3. ↑ Introduction to smooth manifolds - John M Lee - Second Edition
  4. ↑ John M. Lee - Introduction to Smooth Manifolds - second edition
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