Tangent space

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Types of tangent space

Name Symbol Definition Tangent "Vector"
Geometric tangent space Gp(Rn)[1] The set of tangents to a point in Rn

Gp(Rn)={(p,v)|vRn} - the set of all arrows at p

vGp(Rn)v=(u,p) for uRn - pretty much just a vector
Tangent space (to Rn) Tp(Rn) The set of all derivations at p]]

ωTp(Rn)ω:C(Rn)R is a derivation

Tangent vector
Tangent space (to a smooth manifold M) Tp(M) The set of all derivations at p, here a derivation is an R-linear map, ω:C(M)R which satisfies the Leibniz rule. Recall C(M) is the set of all smooth functions on our smooth manifold Tangent vector (to a manifold)
Tangent space (in terms of germs) Dp(M) The set of all derivations of Cp(M) - the set of all germs of smooth functions at a point, that is:

ωDp(M)ω:Cp(M)R is a derivation
See: Set of all derivations of a germ at a point

See

Geometric Tangent Space

The Geometric tangent space to Rn at p[2] is defined as follows:

  • Gp(Rn)={(p,v)|vRn} - the set of all arrows rooted at p

Vector space

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

  • vp+wp=(v+w)p
  • c(vp)=(cv)p

Notations

  • John M Lee uses Rnp to mean the same thing ( Gp(Rn) )

Tangent Space

The Tangent space to Rn at p[3] is defined as follows:

  • Tp(Rn)={ω:ω is a derivation at p} - that is:
    ωTp(Rn)ω:C(Rn)R where

Isomorphism between geometric tangent space and tangent space

Infact the geometric tangent space and tangent space to Rn at p are linearly isomorphic to each other.

Proposition:

  • α:Gp(Rn)Tp(Rn) given by:
    • α:vp[Dv|p:C(Rn)R]
is a linear isomorphism
[Expand]

Theorem: The map α:Gp(Rn)Tp(Rn) given by α:vp[Dv|p:C(Rn)R] is a linear isomorphism


Tangent Space to a Manifold

The tangent space to a manifold M at p is defined as follows:

  • Tp(M)={ω:ω is a derivation at p} - that is:
    ωTp(M)ω:C(M)R where

Recall C(M) is the set of all smooth functions on a smooth manifold M


See also

OLD PAGE

I prefer to denote the tangent space (of a set A at a point p) by Tp(A) - as this involves the letter T for tangent however one author[4] uses Tp(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.

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 Tp(Rn)={(p,v)|vRn}

Generally then we may say: Tp(A)={(p,v)|vA}

Notation

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

Rather than writing (p,v) we may write:

  • v (if it is implicitly understood that this is a tangent to the point p)
  • va
  • v|a

Why ordered pairs

Ordered pairs are used because now the tangent space at two distinct points are disjoint sets, that is αβTα(A)Tβ(A)=

Vector space

Tp(A) is a vector space when equipped with the following definitions:

  • va+wa=(v+w)a
  • c(va)=(cv)a

It is easily seen that the basis for this is the standard basis {e1|p,,en|p} and that the tangent space Tp(A) is basically just a copy of A

See also

References

  1. Jump up Alec's notation - John M Lee uses Rnp and it is distinct from Tp(Rn)
  2. Jump up Introduction to smooth manifolds - John M Lee - Second Edition
  3. Jump up Introduction to smooth manifolds - John M Lee - Second Edition
  4. Jump up John M. Lee - Introduction to Smooth Manifolds - second edition