tangent space

Summary The tangent space of differential manifold $M$ at apoint $x\in M$ is the vector space whose elements are velocities oftrajectories that pass through $x$. The standard notation for thetangent space of $M$ at the point $x$ is $T_{x}M$.

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

$${\gamma }_i:I_i\to M,I_i\subset ℝ,i=1,2$$

betwo differentiable trajectories passing through $x$ at times $t_1\in I_1,t_2\in I_2$, respectively. We say that these trajectories are infirst order contact at $x$ if for all differentiable functions$f:U\to ℝ$ defined in some neighbourhood $U\subset M$ of$x$, we have

$${(f\circ {\gamma }_1)}^{\prime }(t_1)={(f\circ {\gamma }_2)}^{\prime }(t_2).$$

First ordercontact is an equivalence relation, and we define $T_{x}M$, the tangent space of $M$ at $x$, to be the set of correspondingequivalence classes.

Given a trajectory

$$\gamma :I\to M,I\subset ℝ$$

passing through $x$ attime $t\in I$, we define $\dot{\gamma }(t)$ the tangent vector ,a.k.a. the velocity, of $\gamma$ at time $t$, to be theequivalence class of $\gamma$ modulo first order contact. We endow$T_{x}M$ with the structure of a real vector space by identifying itwith ${ℝ}^n$ relative to a system of local coordinates. Theseidentifications will differ from chart to chart, but they will all belinearly compatible.

To describe this identification, consider a coordinate chart

$$\alpha :U_{\alpha }\to {ℝ}^n,U_{\alpha }\subset M,x\in U.$$

We call the real vector

$${(\alpha \circ \gamma )}^{\prime }(t)\in {ℝ}^n$$

the representation of$\dot{\gamma }(t)$ relative to the chart $\alpha$. It is a simple exerciseto show that two trajectories are in first order contact at $x$ if andonly if their velocities have the same representation. Another simpleexercise will show that for every $𝐮\in {ℝ}^n$ the trajectory

$$t\to {\alpha }^{-1}(\alpha (x)+t𝐮)$$

has velocity $𝐮$ relative to the chart $\alpha$. Hence, everyelement of ${ℝ}^n$ represents some actual velocity, and thereforethe mapping $T_{x}M\to {ℝ}^n$ given by

$$[\gamma ]\to {(\alpha \circ \gamma )}^{\prime }(t),\gamma (t)=x,$$

is a bijection.

Finally if $\beta :U_{\beta }\to {ℝ}^n,U_{\beta }\subset M,x\in U_{\beta }$ is another chart, then for all differentiabletrajectories $\gamma (t)=x$ we have

$${(\beta \circ \gamma )}^{\prime }(t)=J{(\alpha \circ \gamma )}^{\prime }(t),$$

where $J$ isthe Jacobian matrix at $\alpha (x)$ of the suitably restricted mapping$\beta \circ {\alpha }^{-1}:\alpha (U_{\alpha }\cap U_{\beta })\to {ℝ}^n$. Thelinearity of the above relationimplies that the vector space structure of $T_{x}M$ is independent of thechoice of coordinate chart.

Definition (Classical). Historically, tangent vectors werespecified as elements of ${ℝ}^n$ relative to some system ofcoordinates, a.k.a. a coordinate chart. This point of view naturallyleads to the definition of a tangent space as ${ℝ}^n$ modulo changesof coordinates.

Let $M$ be a differential manifold represented as a collection ofparameterization domains

$$\{V_{\alpha }\subset {ℝ}^n:\alpha \in 𝒜\}$$

indexed by labelsbelonging to a set $𝒜$, andtransition function diffeomorphisms

$${\sigma }_{\alpha \beta }:V_{\alpha \beta }\to V_{\beta \alpha },\alpha ,\beta \in 𝒜,V_{\alpha \beta }\subset V_{\alpha }$$

Set

$$\widehat{M}=\{(\alpha ,x)\in 𝒜\times {ℝ}^n:x\in V_{\alpha }\},$$

and recall that a points of the manifold are represented by elementsof $\widehat{M}$ modulo an equivalence relation imposed by the transition functions[see Manifold — Definition (Classical)].For a transition function ${\sigma }_{\alpha \beta }$, let

$$\mathrm{J}{\sigma }_{\alpha \beta }:V_{\alpha \beta }\to {Mat}_{n,n}(ℝ)$$

denote the corresponding Jacobian matrix of partial derivatives. Wecall a triple

$$(\alpha ,x,𝐮),\alpha \in 𝒜,x\in V_{\alpha },𝐮\in {ℝ}^n$$

the representation of a tangent vector at $x$ relative to coordinatesystem $\alpha$, and make the identification

$$(\alpha ,x,𝐮)\simeq (\beta ,{\sigma }_{\alpha \beta }(x),[\mathrm{J}{\sigma }_{\alpha \beta }](x)(𝐮)),\alpha ,\beta \in 𝒜,x\in V_{\alpha \beta },𝐮\in {ℝ}^n.$$

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

Notes. The notion of tangent space derives from the observationthat there is no natural way to relate and compare velocities atdifferent points of a manifold. This is already evident when weconsider objects moving on a surface in 3-space, where the velocitiestake their value in the tangent planes of the surface. On a generalsurface, distinct points correspond to distinct tangent planes, andtherefore the velocities at distinct points are not commensurate.

The situation is even more complicated for an abstract manifold, whereabsent an ambient Euclidean setting there is, apriori, no obvious“tangent plane” where the velocities can reside. This point of viewleads to the definition of a velocity as some sort of equivalenceclass.

See also: tangent bundle, connection, parallel translation