Tangent space
Contents
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
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
- [ilmath]\omega[/ilmath] is [ilmath]\mathbb{R} [/ilmath]-linear
- [ilmath]\omega[/ilmath] satisfies the Leibniz rule
- [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
- [ilmath]\omega[/ilmath] is [ilmath]\mathbb{R} [/ilmath]-linear
- [ilmath]\omega[/ilmath] satisfies the Leibniz rule
- [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
- 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^{[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
- ↑ 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]
- ↑ Introduction to smooth manifolds - John M Lee - Second Edition
- ↑ Introduction to smooth manifolds - John M Lee - Second Edition
- ↑ John M. Lee - Introduction to Smooth Manifolds - second edition