tangent space

Summary The tangent spacePlanetmathPlanetmath of differential manifold M at a point xM is the vector spaceMathworldPlanetmath whose elements are velocities of trajectories that pass through x. The standard notation for the tangent space of M at the point x is TxM.

Definition (Standard). Let M be a differential manifold and x a point of M. Let


be two differentiableMathworldPlanetmathPlanetmath trajectories passing through x at times t1I1,t2I2, respectively. We say that these trajectories are in first order contact at x if for all differentiable functions f:U defined in some neighbourhood UM of x, we have


First order contact is an equivalence relationMathworldPlanetmath, and we define TxM, the tangent space of M at x, to be the set of corresponding equivalence classesMathworldPlanetmath.

Given a trajectory


passing through x at time tI, 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 endow TxM with the structureMathworldPlanetmath of a real vector space by identifying it with n 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 of γ˙(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 𝐮n the trajectory


has velocity 𝐮 relative to the chart α. Hence, every element of n represents some actual velocity, and therefore the mapping TxMn given by


is a bijectionMathworldPlanetmath.

Finally if β:Uβn,UβM,xUβ 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 mapping βα-1:α(UαUβ)n. The linearity of the above relationMathworldPlanetmath implies that the vector space structure of TxM is independent of the choice of coordinate chart.

Definition (Classical). Historically, tangent vectors were specified as elements of n relative to some system of coordinatesMathworldPlanetmathPlanetmath, a.k.a. a coordinate chart. This point of view naturally leads to the definition of a tangent space as n 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, and transition function diffeomorphisms




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 σαβ, 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


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

Title tangent space
Canonical name TangentSpace
Date of creation 2013-03-22 12:21:04
Last modified on 2013-03-22 12:21:04
Owner rmilson (146)
Last modified by rmilson (146)
Numerical id 5
Author rmilson (146)
Entry type Definition
Classification msc 53-00
Synonym tangent vector
Related topic Manifold