Summary The tangent space of differential manifold at a point is the vector space whose elements are velocities of trajectories that pass through . The standard notation for the tangent space of at the point is .
Definition (Standard). Let be a differential manifold and a point of . Let
be two differentiable trajectories passing through at times , respectively. We say that these trajectories are in first order contact at if for all differentiable functions defined in some neighbourhood of , we have
First order contact is an equivalence relation, and we define , the tangent space of at , to be the set of corresponding equivalence classes.
Given a trajectory
passing through at time , we define the tangent vector , a.k.a. the velocity, of at time , to be the equivalence class of modulo first order contact. We endow with the structure of a real vector space by identifying it with relative to a system of local coordinates. These identifications will differ from chart to chart, but they will all be linearly compatible.
To describe this identification, consider a coordinate chart
We call the real vector
the representation of relative to the chart . It is a simple exercise to show that two trajectories are in first order contact at if and only if their velocities have the same representation. Another simple exercise will show that for every the trajectory
has velocity relative to the chart . Hence, every element of represents some actual velocity, and therefore the mapping given by
is a bijection.
Finally if is another chart, then for all differentiable trajectories we have
where is the Jacobian matrix at of the suitably restricted mapping . The linearity of the above relation implies that the vector space structure of is independent of the choice of coordinate chart.
Definition (Classical). Historically, tangent vectors were specified as elements of relative to some system of coordinates, a.k.a. a coordinate chart. This point of view naturally leads to the definition of a tangent space as modulo changes of coordinates.
Let be a differential manifold represented as a collection of parameterization domains
indexed by labels belonging to a set , and transition function diffeomorphisms
and recall that a points of the manifold are represented by elements of modulo an equivalence relation imposed by the transition functions [see Manifold — Definition (Classical)]. For a transition function , let
denote the corresponding Jacobian matrix of partial derivatives. We call a triple
the representation of a tangent vector at relative to coordinate system , and make the identification
to arrive at the definition of a tangent vector at .
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 Euclidean 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, connection, parallel translation
|Date of creation||2013-03-22 12:21:04|
|Last modified on||2013-03-22 12:21:04|
|Last modified by||rmilson (146)|