tangent space (original) (raw)

Given a trajectory

passing through x at time t∈I, we define γ˙⁢(t) the tangent vector , a.k.a. the velocity, of γ at time t, to be the equivalence class of γ modulo first order contact. We endowTx⁢M with the structureMathworldPlanetmath of a real vector space by identifying it with ℝn relative to a system of local coordinates. These identifications will differ from chart to chart, but they will all be linearly compatibleMathworldPlanetmath.

To describe this identification, consider a coordinate chart

We call the real vector

the representation ofγ˙⁢(t) relative to the chart α. It is a simple exercise to show that two trajectories are in first order contact at x if and only if their velocities have the same representation. Another simple exercise will show that for every 𝐮∈ℝn the trajectory

has velocity 𝐮 relative to the chart α. Hence, every element of ℝn represents some actual velocity, and therefore the mapping Tx⁢M→ℝn given by

is a bijectionMathworldPlanetmath.

Finally if β:Uβ→ℝn,Uβ⊂M,x∈Uβ is another chart, then for all differentiable trajectories γ⁢(t)=x we have

where J is the Jacobian matrix at α⁢(x) of the suitably restricted mappingβ∘α-1:α⁢(Uα∩Uβ)→ℝn. The linearity of the above relationMathworldPlanetmathimplies that the vector space structure of Tx⁢M is independent of the choice of coordinate chart.

Definition (Classical). Historically, tangent vectors were specified as elements of ℝn relative to some system ofcoordinatesMathworldPlanetmathPlanetmath, a.k.a. a coordinate chart. This point of view naturally leads to the definition of a tangent space as ℝn modulo changes of coordinates.

Let M be a differential manifold represented as a collectionMathworldPlanetmath of parameterization domains

indexed by labels belonging to a set 𝒜, andtransition function diffeomorphisms

σα⁢β:Vα⁢β→Vβ⁢α,α,β∈𝒜,Vα⁢β⊂Vα

Set

and recall that a points of the manifold are represented by elements of M^ modulo an equivalence relation imposed by the transition functions [see Manifold — Definition (Classical)]. For a transition function σα⁢β, let

denote the corresponding Jacobian matrix of partial derivativesMathworldPlanetmath. We call a triple

the representation of a tangent vector at x relative to coordinate systemMathworldPlanetmath α, and make the identification

(α,x,𝐮)≃(β,σα⁢β⁢(x),[J⁢σα⁢β]⁢(x)⁢(𝐮)),α,β∈𝒜,x∈Vα⁢β,𝐮∈ℝn.

to arrive at the definition of a tangent vector at x.

Notes. The notion of tangent space derives from the observation that there is no natural way to relate and compare velocities at different points of a manifold. This is already evident when we consider objects moving on a surface in 3-space, where the velocities take their value in the tangent planes of the surface. On a general surface, distinct points correspond to distinct tangent planes, and therefore the velocities at distinct points are not commensurate.

The situation is even more complicated for an abstract manifold, where absent an ambient EuclideanPlanetmathPlanetmath setting there is, apriori, no obvious “tangent plane” where the velocities can reside. This point of view leads to the definition of a velocity as some sort of equivalence class.

See also: tangent bundle, connectionMathworldPlanetmath, parallel translation