nLab
tangent bundle

Context

Differential geometry

differential geometry

synthetic differential geometry

Axiomatics

Models

Concepts

Theorems

Applications

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Contents

Idea

The tangent bundle TXXT X \to X of a (sufficiently differentiable) space XX is a bundle over XX whose fiber over a point xXx \in X is the tangent space at that point, namely the collection of infinitesimal curves in XX emanating at xx: “tangent vectors”.

For nice enough spaces such as differentiable manifolds or more generally microlinear spaces, the tangent bundle of XX is a vector bundle over XX.

For example the graphics on the right shows the 2-sphere with one of its tangent spaces. The tangent bundle of the sphere is the union of all these tangent spaces, regarded as a topological bundle of vector space (a vector bundle) over the 2-sphere.

graphics grabbed from Hatcher

With a notion of tangent bundle comes the following terminology

  • A tangent vector on XX at xXx \in X is an element of T xXT_x X.

  • The tangent space of XX at a point xx is the fiber T x(X)T_x(X) of T *(X)T_*(X) over xx;.

  • A tangent vector field on XX is a section of TXT X.

The precise definition of tangent bundle depends on the nature of the ambient category of spaces. Below we give first the traditional definitions in ordinary differential geometry. Then we discuss the construction in more general context of smooth toposes in synthetic differential geometry and other categories of generalized smooth spaces.

Definitions

Traditional definition

We discuss the tangent bundle of a differentiable manifold by first defining tangent vectors as equivalence classes of differentiable curves in the manifold, then analyzing this construction locally over an atlas, and then gluing these local constructions together via transition functions.

Definition

(tangency relation on differentiable curves)

Let XX be a differentiable manifold of dimension nn and let xXx \in X be a point. On the set of smooth functions of the form

γ: 1X \gamma \;\colon\; \mathbb{R}^1 \longrightarrow X

such that

γ(0)=x \gamma(0) = x

define the relations

(γ 1γ 2) nϕchartU iXU i{x}(ddt(ϕ 1γ 1)(0)=ddt(ϕ 1γ 2)(0)) (\gamma_1 \sim \gamma_2) \coloneqq \underset{ { { \mathbb{R}^n \underoverset{}{\phi \, \text{chart}}{\to} U_i \subset X } } \atop { U_i \supset \{x\} } }{ \exists } \left( \frac{d}{d t}(\phi^{-1}\circ \gamma_1)(0) = \frac{d}{d t}(\phi^{-1}\circ \gamma_2)(0) \right)

and

(γ 1 γ 2) nϕchartU iXU i{x}(ddt(ϕ 1γ 1)(0)=ddt(ϕ 1γ 2)(0)) (\gamma_1 {\sim^\prime} \gamma_2) \coloneqq \underset{ { { \mathbb{R}^n \underoverset{}{\phi \, \text{chart}}{\to} U_i \subset X } } \atop { U_i \supset \{x\} } }{ \forall } \left( \frac{d}{d t}(\phi^{-1}\circ \gamma_1)(0) = \frac{d}{d t}(\phi^{-1}\circ \gamma_2)(0) \right)

saying that two such functions are related precisely if either there exists a chart around xx such that (or else for all charts around xx it is true that) the first derivative of the two functions regarded via the given chart as functions 1 n\mathbb{R}^1 \to \mathbb{R}^n, coincide at t=0t = 0 (with tt denoting the canonical coordinate function on \mathbb{R}).

Lemma

(tangency is equivalence relation)

The two relations in def. 1 are equivalence relations and they coincide.

Proof

First to see that they conincide, we need to show that if the derivatives in question coincide in one chart nϕU iX\mathbb{R}^n \underoverset{\simeq}{\phi}{\to} U_i \subset X, that then they coincide also in any other chart nψU jX\mathbb{R}^n \underoverset{\simeq}{\psi}{\to} U_j \subset X.

Write

U ijU iU j U_{i j} \coloneqq U_i \cap U_j

for the intersection of the two charts.

First of all, since the derivative may be computed in any open neighbourhood around t=0t = 0, and since the differentiable functions γ i\gamma_i are in particular continuous functions, we may restrict to the open neighbourhood

Vγ 1 1(U ij)γ 2 1(U ij) V \coloneqq \gamma_1^{-1}( U_{i j} ) \cap \gamma_2^{-1}(U_{i j}) \;\subset\; \mathbb{R}

of 00 \in \mathbb{R} and consider the derivatives of the functions

γ i ϕ(ϕ| U ijγ i| V):Vϕ 1(U ij) n \gamma_i^{\phi} \;\coloneqq\; (\phi\vert_{U_{i j}} \circ \gamma_i\vert_{V}) \;\colon\; V \longrightarrow \phi^{-1}(U_{i j}) \subset \mathbb{R}^n

and

γ i ψ(ψ| U ijγ i| V):Vψ 1(U ij) n. \gamma_i^{\psi} \;\coloneqq\; (\psi\vert_{U_{i j}} \circ \gamma_i \vert _{V}) \;\colon\; V \longrightarrow \psi^{-1}(U_{i j}) \subset \mathbb{R}^n \,.

But then by definition of the differentiable atlas, there is the differentiable function

αϕ 1(U ij)ϕU ijψ 1ψ 1(U ij) \alpha \;\coloneqq\; \phi^{-1}(U_{i j}) \underoverset{\simeq}{\phi}{\longrightarrow} U_{i j} \underoverset{\simeq}{\psi^{-1}}{\longrightarrow} \psi^{-1}(U_i j)

such that

γ i ψ=αγ i ϕ \gamma_i^\psi = \alpha \circ \gamma_i^\phi

for i{1,2}i \in \{1,2\}. The chain rule now relates the derivatives of these functions as

ddtγ i ψ=(Dα)(ddtγ i ϕ). \frac{d}{d t} \gamma_i^\psi \;=\; (D \alpha) \circ \left(\frac{d}{d t} \gamma_i^\phi\right) \,.

Since α\alpha is a diffeomorphism and since derivatives of diffeomorphisms are linear isomorphisms, this says that the derivative of γ i ϕ\gamma_i^\phi is related to that of γ i ψ\gamma_i^\psi by a linear isomorphism, and hence

(ddt(γ 1) ϕ=ddt(γ 2 ϕ))(ddt(γ 1) ψ=ddt(γ 2 ψ)). \left( \frac{d}{d t}(\gamma_1)^\phi = \frac{d}{d t}(\gamma_2^\phi) \right) \;\Leftrightarrow\; \left( \frac{d}{d t}(\gamma_1)^\psi = \frac{d}{d t}(\gamma_2^\psi) \right) \,.

Finally, that either relation is an equivalence relation is immediate.

Definition

(tangent vector)

Let XX be a differentiable manifold and xXx \in X a point. Then a tangent vector on XX at xx is an equivalence class of the the tangency equivalence relation (def. 1, lemma 1).

The set of all tangent vectors at xXx \in X is denoted T xXT_x X.

Lemma

(real vector space structure on tangent vectors)

For XX a differentiable manifold of dimension nn and xXx \in X any point, let nϕUX\mathbb{R}^n \underoverset{\simeq}{\phi}{\to} U \subset X be a chart with xUXx \in U \subset X.

Then there is induced a bijection of sets

nT xX \mathbb{R}^n \overset{\simeq}{\longrightarrow} T_x X

from the nn-dimensional Cartesian space to the set of tangent vectors at xx (def. 2) given by sending v n\vec v \in \mathbb{R}^n to the equivalence class of the following differentiable curve:

γ v ϕ 1 ()v n ϕ U iX t AAA tv AAA ϕ(ϕ 1(x)+tv). \array{ \gamma^\phi_{\vec v} & \mathbb{R}^1 &\overset{ (-)\cdot \vec v }{\longrightarrow}& \mathbb{R}^n &\underoverset{\simeq}{\phi}{\longrightarrow}& U_i \subset X \\ & t &\overset{\phantom{AAA}}{\mapsto}& t \vec v &\overset{\phantom{AAA}}{\mapsto}& \phi(\phi^{-1}(x) + t \vec v) } \,.

For nϕUX\mathbb{R}^n \underoverset{\simeq}{\phi'}{\longrightarrow} U' \subset X another chart with xUXx \in U' \subset X, then the linear isomorphism relating these two identifications is the derivative

d((ϕ) 1ϕ) ϕ 1(x)GL(n,) d \left((\phi')^{-1} \circ \phi \right)_{ \phi^{-1}(x) } \in GL(n,\mathbb{R})

of the gluing function of the two charts at the point xx:

n d((ϕ) 1ϕ)(x) n T xX. \array{ \mathbb{R}^n && \overset{ d \left((\phi')^{-1} \circ \phi\right) (x)}{\longrightarrow} && \mathbb{R}^n \\ & {}_{\mathllap{\simeq}}\searrow && \swarrow_{\mathrlap{\simeq}} \\ && T_x X } \,.

This is also called the transition function between the two local identifications of the tangent space.

If { nϕ iU iX} iI\left\{ \mathbb{R}^n \underoverset{\simeq}{\phi_i}{\to} U_i \subset X \right\}_{i \in I} is an atlas of the differentiable manifold XX, then the transition functions

{g ijd(ϕ j 1ϕ i) ϕ 1():U iU jGL(n,)} i,jI \left\{ g_{i j} \coloneqq d( \phi_j^{-1} \circ \phi_i )_{\phi^{-1}(-)} \colon U_i \cap U_j \longrightarrow GL(n,\mathbb{R}) \right\}_{i,j \in I}

defined this way satisfy the following Cech cocycle conditions for all i,jIi,j \in I, xU iU jx \in U_i \cap U_j

  1. g ii(x)=id ng_{i i}(x) = id_{\mathbb{R}^n};

  2. g jkg ij(x)=g ik(x)g_{j k}\circ g_{i j}(x) = g_{i k}(x).

Proof

The bijectivity of the map is immediate from the fact that the first derivative of ϕ 1γ v ϕ\phi^{-1}\circ \gamma^\phi_{\vec v} at ϕ 1(x)\phi^{-1}(x) is v\vec v.

The formula for the transition function now follows with the chain rule:

d((ϕ) 1ϕ(ϕ 1(x)()v)) 0=d((ϕ) 1ϕ) ϕ 1(x)d(ϕ 1(x)+()v) 0v. d \left( (\phi')^{-1} \circ \phi( \phi^{-1}(x) (-) \vec v ) \right)_0 = d \left( (\phi')^{-1} \circ \phi \right)_{\phi^{-1}(x)} \circ \underset{ \vec v }{\underbrace{ d ( \phi^{-1}(x) +(-)\vec v )_0 }} \,.

Similarly the Cech cocycle condition follows by the chain rule:

g jkg ij(x) =d(ϕ k 1ϕ j) ϕ j 1(x)d(ϕ j 1ϕ i) ϕ i 1(x) =d(ϕ k 1ϕ jϕ j 1ϕ i) ϕ i 1(x) =d(ϕ k 1ϕ i) ϕ i 1(x) =g ik(x). \begin{aligned} g_{j k} \circ g_{i j}(x) & = d( \phi_k^{-1} \circ \phi_j )_{\phi_j^{-1}(x)} \circ d( \phi_j^{-1} \circ \phi_i )_{\phi_i^{-1}(x)} \\ & = d( \phi_k^{-1} \circ \phi_j \circ \phi_j^{-1} \circ \phi_i )_{\phi_i^{-1}(x)} \\ & = d( \phi_k^{-1} \circ \phi_i )_{\phi_i^{-1}(x)} \\ &= g_{i k}(x) \end{aligned} \,.
Definition

(tangent space)

For XX a differentiable manifold and xXx \in X a point, then the tangent space of XX at xx is the set T xXT_x X of tangent vectors at xx (def. 2) regarded as a real vector space via lemma 2.

Example

(tangent bundle of Euclidean space)

If X= nX = \mathbb{R}^n is itself a Euclidean space, then for any two points x,yXx,y \in X the tangent spaces T xXT_x X and T yXT_y X (def. 3) are canonically identified with each other:

Using the vector space (or just affine space) structure of X= nX = \mathbb{R}^n we may send every smooth function γ:X\gamma \colon \mathbb{R} \to X to the smooth function

γ:tγ(t)+(xy). \gamma' \colon t \mapsto \gamma(t) + (x-y) \,.

This gives a linear bijection

ϕ x,y:T xXT yX \phi_{x,y} \colon T_x X \overset{\simeq}{\longrightarrow} T_y X

and these linear bijections are compatible, in that for x,y,z nx,y,z \in \mathbb{R}^n any three points, then

ϕ y,zϕ x,y=ϕ x,z:T xXT yY. \phi_{y,z} \circ \phi_{x,y} = \phi_{x,z} \;\colon\; T_x X \longrightarrow T_y Y \,.

Moreover, by lemma 2, each tangent space is identified with n\mathbb{R}^n itself, and this identification in turn is compatible with all the above identifications:

n T xX ϕ x,y T yY. \array{ && \mathbb{R}^n \\ & {}^{\mathllap{\simeq}}\swarrow && \searrow^{\mathrlap{\simeq}} \\ T_x X && \underoverset{\phi_{x,y}}{\simeq}{\longrightarrow} && T_y Y } \,.

Therefore it makes sense to canonically identify all the tangent spaces of Euclidean space with that Euclidean space itself. As a result, the collection of all the tangent spaces of Euclidean space is naturally identified with the Cartesian product

T n= n× n T \mathbb{R}^n = \mathbb{R}^n \times \mathbb{R}^n

equipped with the projection on the first factor

T n= n× n π=pr 1 n, \array{ T \mathbb{R}^n = \mathbb{R}^n \times \mathbb{R}^n \\ \downarrow^{\mathrlap{\pi = pr_1}} \\ \mathbb{R}^n } \,,

because then the pre-image of a singleton {x} n\{x\} \subset \mathbb{R}^n under this projection are canonically identified with the above tangent spaces:

π 1({x})T x n. \pi^{-1}(\{x\}) \simeq T_x \mathbb{R}^n \,.

This way, if we equip T n= n× nT \mathbb{R}^n = \mathbb{R}^n \times \mathbb{R}^n with the product space topology, then T nπ nT \mathbb{R}^n \overset{\pi}{\longrightarrow} \mathbb{R}^n becomes a trivial topological vector bundle.

This is called the tangent bundle of the Euclidean space n\mathbb{R}^n regarded as a differentiable manifold.

Remark

(chain rule is functoriality of tangent space construction on Euclidean spaces)

Consider the assignment that sends

  1. every Euclidean space n\mathbb{R}^n to its tangent bundle T nT \mathbb{R}^n according to def. 1;

  2. every differentiable function f: n 1 n 2f \colon \mathbb{R}^{n_1} \longrightarrow \mathbb{R}^{n_2} to the function on tangent vectors (def. 2) induced by postcomposition with ff

    T n 1 f() T n 2 [ 1γ n 1] [ 1fγ n 2] \array{ T \mathbb{R}^{n_1} &\overset{f \circ (-)}{\longrightarrow}& T \mathbb{R}^{n_2} \\ \left[ \mathbb{R}^1 \overset{\gamma}{\longrightarrow} \mathbb{R}^{n_1} \right] &\mapsto& \left[ \mathbb{R}^1 \overset{ f \circ \gamma}{\longrightarrow} \mathbb{R}^{n_2} \right] }

By the chain rule we have that the derivative of the composite curve fγf \circ \gamma is

d(fγ) t=(df γ(x))dγ d (f \circ \gamma)_t = (d f_{\gamma(x)}) \circ d\gamma

and hence that under the identification T n n× nT \mathbb{R}^n \simeq \mathbb{R}^n \times \mathbb{R}^n of example 1 this assignment takes ff to its derivative

n 1× n 1 df) n 2× n 2 (x,v) (f(x),df x(v)), \array{ \mathbb{R}^{n_1} \times \mathbb{R}^{n_1} &\overset{ d f )}{\longrightarrow}& \mathbb{R}^{n_2} \times \mathbb{R}^{n_2} \\ (x,\vec v) &\mapsto& (f(x), d f_x(\vec v)) } \,,

Conversely, in the first form above the assignment ff()f \mapsto f \circ (-) manifestly respects composition (and identity functions). Viewed from the second perspective this respect for composition is once again the chain rule d(gf)=(df)(dg)d(g \circ f) = (d f)\circ (d g):

Y f g X gf ZAAAAAA TY df dg TX d(gf) TZ. \array{ && Y \\ & {}^{\mathllap{f}}\nearrow && \searrow^{\mathrlap{g}} \\ X && \underset{ g\circ f}{\longrightarrow} && Z } \phantom{AAA} \mapsto \phantom{AAA} \array{ && T Y \\ & {}^{\mathllap{d f}}\nearrow && \searrow^{\mathrlap{d g}} \\ T X && \underset{d(g \circ f)}{\longrightarrow} && T Z } \,.

In the language of category theory this says that the assignment

CartSp T CartSp X TX f df Y TY \array{ CartSp &\overset{T}{\longrightarrow}& CartSp \\ X &\mapsto& T X \\ {}^{\mathllap{ f }}\downarrow && \downarrow^{\mathrlap{d f}} \\ Y &\mapsto& T Y }

is an endofunctor on the category CartSp whose

  1. objects are the Euclidean spaces n\mathbb{R}^n for nn \in \mathbb{N};

  2. morphisms are the differentiable functions between these (for any chosen differentiability class C kC^k with k>0k \gt 0).

We may now globalize the tangent bundle of Euclidean space to tangent bundles of general differentiable manifolds:

Definition

(tangent bundle of a differentiable manifold)

Let XX be a differentiable manifold with atlas { nϕ iU iX} iI\left\{ \mathbb{R}^n \underoverset{\simeq}{\phi_i}{\to} U_i \subset X\right\}_{i \in I}.

Equip the set of all tangent vectors (def. 2), i.e. the disjoint union of the sets of tangent vectors

TXxXT xXAAAas underlying sets T X \;\coloneqq\; \underset{x \in X}{\sqcup} T_x X \phantom{AAA} \text{as underlying sets}

with a topology τ TX\tau_{T X} by declaring a subset UTXU \subset T X to be an open subset precisely if for all charts nϕ iU iX\mathbb{R}^n \underoverset{\simeq}{\phi_i}{\to} U_i \subset X we have that its preimage under

2n n× n dϕ TX (x,v) AAA [tϕ(x+tvectv)] \array{ \mathbb{R}^{2n} \simeq \mathbb{R}^n \times \mathbb{R}^n & \overset{d \phi}{\longrightarrow} & T X \\ (x, \vec v) &\overset{\phantom{AAA}}{\mapsto}& [ t \mapsto \phi(x + t \vect v) ] }

is open in the Euclidean space 2n\mathbb{R}^{2n} with its metric topology.

Equipped with the function

TX AAp xAA X (x,v) AAAA x \array{ T X &\overset{\phantom{AA}p_x \phantom{AA}}{\longrightarrow}& X \\ (x,v) &\overset{\phantom{AAAA}}{\mapsto}& x }

this is called the tangent bundle of XX.

Equivalently this means that the tangent bundle TXT X is the topological vector bundle which is glued (via this example) from the transition functions g ijd(ϕ j 1ϕ i) ϕ 1()g_{i j} \coloneqq d(\phi_j^{-1} \circ \phi_i)_{\phi^{-1}(-)} from lemma 2:

TX(iIU i× n)/({d(ϕ j 1ϕ i)} i,jI). T X \;\coloneqq\; \left( \underset{i \in I}{\sqcup} U_i \times \mathbb{R}^n \right)/\left( \left\{ d( \phi_j^{-1} \circ \phi_i ) \right\}_{i, j \in I} \right) \,.

(Notice that, by examples 1, each U i× nTU iU_i \times \mathbb{R}^n \simeq T U_i is the tangent bundle of the chart U i nU_i \simeq \mathbb{R}^n.)

The co-projection maps of this quotient topological space construction constitute an atlas

{ 2nTU iTX} iI. \left\{ \mathbb{R}^{2n} \underoverset{\simeq}{}{\to} T U_i \subset T X \right\}_{i \in I} \,.
Lemma

(tangent bundle is differentiable vector bundle)

If XX is a (p+1)(p+1)-times differentiable manifold, then the total space of the tangent bundle def. 4 is a pp-times differentiable manifold in that

  1. (TX,τ TX)(T X, \tau_{T X}) is a paracompact Hausdorff space;

  2. The gluing functions of the atlas { 2ndϕ iTU iTX} iI\left\{ \mathbb{R}^{2n} \underoverset{\simeq}{d \phi_i}{\to} T U_i \subset T X \right\}_{i \in I} are pp-times continuously differentiable.

Moreover, the projection π:TXX\pi \colon T X \to X is a pp-times continuously differentiable function.

In summary this makes TXXT X \to X a differentiable vector bundle.

Proof

First to see that TXT X is Hausdorff:

Let (x,v),(x,v)TX(x,\vec v), (x', \vec v') \in T X be two distinct points. We need to product disjoint openneighbourhoods of these points in TXT X. Since in particular x,xXx,x' \in X are distinct, and since XX is Hausdorff, there exist disjoint open neighbourhoods U x{x}U_x \supset \{x\} and U x{x}U_{x'} \supset \{x'\}. Their pre-images π 1(U x)\pi^{-1}(U_x) and π 1(U x)\pi^{-1}(U_{x'}) are disjoint open neighbourhoods of (x,v)(x,\vec v) and (x,vectv)(x',\vect v'), respectively.

Now to see that TXT X is paracompact.

Let {U iTX} iI\{U_i \subset T X\}_{i \in I} be an open cover. We need to find a locally finite refinement. Notice that π:TXX\pi \colon T X \to X is an open map (by this example) so that {π(U i)X} iI\{\pi(U_i) \subset X\}_{i \in I} is an open cover of XX.

Let now { nϕ jV jX} jJ\{\mathbb{R}^n \underoverset{\simeq}{\phi_j}{\to} V_j \subset X\}_{j \in J} be an atlas for XX and consider the open common refinement

{π(U i)V jX} iI,jJ. \left\{ \pi(U_i) \cap V_j \subset X \right\}_{i \in I, j \in J} \,.

Since this is still an open cover of XX and since XX is paracompact, this has a locally finite refinement

{V jX} jJ \left\{ V'_{j'} \subset X\right\}_{j' \in J'}

Notice that for each jJj' \in J' the product topological space V j× n 2nV'_{j'} \times \mathbb{R}^n \subset \mathbb{R}^{2n} is paracompact (as a topological subspace of Euclidean space it is itself locally compact and second countable and since locally compact and second-countable spaces are paracompact). Therefore the cover

{π 1(V j)U iV j× n} (i,j)I×J \{ \pi^{-1}(V'_{j'}) \cap U_i \subset V'_{j'} \times \mathbb{R}^n \}_{(i,j') \in I \times J'}

has a locally finite refinement

{W k jV j× n} k jK j. \{W_{k_{j'}} \subset V'_{j'} \times \mathbb{R}^n \}_{k_{j'} \in K_{j'}} \,.

We claim now that

{W k jTX} jJ,k jK j \{ W_{k_{j'}} \subset T X \}_{j' \in J', k_{j'} \in K_{j'}}

is a locally finite refinement of the original cover. That this is an open cover refining the original one is clear. We need to see that it is locally finite.

So let (x,v)TX(x,\vec v) \in T X. By local finiteness of {V jX} jJ\{ V'_{j'} \subset X\}_{j' \in J'} there is an open neighbourhood V x{x}V_x \supset \{x\} which intersects only finitely many of the V jXV'_{j'} \subset X. Then by local finiteness of {W k jV j }\{ W_{k_{j'}} \subset V'_{j_'}\}, for each such jj' the point (x,v)(x,\vec v) regarded in V j× nV'_{j'} \times \mathbb{R}^n has an open neighbourhood U jU_{j'} that intersects only finitely many of the W k jW_{k_{j'}}. Hence the intersection π 1(V x)(jU j)\pi^{-1}(V_x) \cap \left( \underset{j'}{\cap} U_{j'} \right) is a finite intersection of open subsets, hence still open, and by construction it intersects still only a finite number of the W k jW_{k_{j'}}.

This shows that TXT X is paracompact.

Finally the statement about the differentiability of the glung functions and of the projections is immediate from the definitions

Proposition

(differentials of differentiable functions between differentiable manifolds)

Let XX and YY be differentiable manifolds and let f:XYf \;\colon\; X \longrightarrow Y be a differentiable function. Then the operation of postcomposition which takes differentiable curves in XX to differentiable curves in YY

Hom Diff( 1,X) f() Hom Diff( 1,Y) ( 1γX) AAA ( 1fγY) \array{ Hom_{Diff}(\mathbb{R}^1, X) &\overset{f \circ (-)}{\longrightarrow}& Hom_{Diff}(\mathbb{R}^1, Y) \\ \left( \mathbb{R}^1 \overset{\gamma}{\to} X \right) &\overset{\phantom{AAA}}{\mapsto}& \left( \mathbb{R}^1 \overset{f \circ \gamma}{\to} Y \right) }

descends at each point xXx \in X to the tangency equivalence relation (def. 1, lemma 1) to yield a function on sets of tangent vectors (def. 2), called the differential df xd f_x of ff at xx

df| x:T xXT f(x)Y. d f|_{x} \;\colon\; T_x X \longrightarrow T_{f(x)} Y \,.

Moreover:

  1. (linear dependence on the tangent vector) these differentials are linear functions with respect to the vector space structure on the tangent spaces from lemma 2, def. 3;

  2. (differentiable dependence on the base point) globally they yield a homomorphism of real differentiable vector bundles between the tangent bundles (def. 4, lemma 3), called the global differential dfd f of ff

    df:TXTY. d f \;\colon\; T X \longrightarrow T Y \,.
  3. (chain rule) The assignment fdff \mapsto d f respects composition in that for XX, YY, ZZ three differentiable manifolds and for

    XAfAYAgAZ X \overset{\phantom{A}f\phantom{A}}{\longrightarrow} Y \overset{\phantom{A}g\phantom{A}}{\longrightarrow} Z

    two composable differentiable functions then their differentials satisfy

    d(gf)=(dg)(df). d(g \circ f) = (d g) \circ (d f) \,.
Proof

All statements are to be tested on charts of an atlas for XX and for YY. On these charts the statement reduces to that of example 1.

Remark

In the language of category theory the statement of prop. 1 says that forming tangent bundles TXT X of differentiable manifolds XX and differentials dfd f of differentiable functions f:XYf \colon X \to Y constitutes a functor

T:DiffVect(Diff) T \;\colon\; Diff \longrightarrow Vect(Diff)

from the category Diff of differentiable manifolds to the category of differentiable real vector bundles.

Definition

(vector field)

Let XX be a differentiable manifold with differentiable tangent bundle TXXT X \to X (def. 4).

A differentiable section v:XTXv \colon X \to T X of the tangent bundle is called a (differentiable) vector field on XX.

Remark

(derivations of smooth functions are vector fields)

Let XX be a smooth manifold and write C (X)C^\infty(X) for the associative algebra over the real numbers of smooth functions XX \longrightarrow \mathbb{R}.

Then every smooth vector field vΓ X(TX)v \in \Gamma_X(T X) (def. 5) induces a function

v:C (X)C (X) \partial_v \;\colon\; C^\infty(X) \longrightarrow C^\infty(X)

by

f:xddt(fγ v x) 0 \partial f \;\colon\; x \mapsto \frac{d}{d t} (f \circ \gamma_{v_x})_0

where γ v x: 1X\gamma_{v_x} \colon \mathbb{R}^1 \to X is a smooth curve which represents the tangent vector v(x)T xXv(x) \in T_x X according to def. 2.

The linearity of derivatives and the product rule of differentiation imply that this function v\partial_v is a derivation on the algebra of smooth functions. Hence there is a function

Γ X(TX) Def(C (X)) v v. \array{ \Gamma_X(T X) &\longrightarrow& Def(C^\infty(X)) \\ v &\mapsto& \partial_v } \,.

It turns out that this function is in fact a bijection: every derivation of the algebra of smooth functions on a smooth manifold arises uniquely from a smooth tangent vector in this way.

For more on this see at derivations of smooth functions are vector fields.

Remark

(notation for tangent vectors in a chart)

Under the bijection of lemma 2 one often denotes the tangent vector corresponding to the the ii-th canonical basis vector of n\mathbb{R}^n by

x iAAor just AA i \frac{\partial}{\partial x^i} \phantom{AA} \text{or just } \phantom{AA} \partial_i

because under the identification of tangent vectors with derivations on the algebra of differentiable functions on XX as above then it acts as the operation of taking the iith partial derivative. The general tangent vector corresponding to v nv \in \mathbb{R}^n is then denoted by

i=1nv ix iAAor just AAi=1nv i i. \underoverset{i = 1}{n}{\sum} v^i \frac{\partial}{\partial x^i} \phantom{AA} \text{or just } \phantom{AA} \underoverset{i = 1}{n}{\sum} v^i \partial_i \,.

Notice that this identification depends on the choice of chart, which is left implicit in this notation.

Sometimes, notably in texts on thermodynamics, one augments this notation to indicate the chart being used by listing the remaining coordinate functions as subscripts. For instance if two functions f,gf,g on a 2-dimensional manifold are used as coordinate functions for a local chart (i.e. so that x 1=fx^1 = f and x 2=gx^2 = g ), then one writes

(/f) gAA(/g) f (\partial/\partial f)_g \phantom{AA} (\partial/\partial g)_f

for the tangent vectors x 1\frac{\partial}{\partial x^1} and x 2\frac{\partial}{\partial x^2}, respectively.

In synthetic differential geometry

The above definitions in ordinary differential geometry suggest the slogan

Tangent vectors are infinitesimal curves in a space.

One of the central motivations for synthetic differential geometry is the desire to provide a context in which this slogan becomes literally formally true.

Definition

(tangent bundle in smooth toposes)

Let (𝒯,(R,+,))(\mathcal{T},(R,+,\cdot)) be a smooth topos and write D={ϵR|ϵ 2=0}D = \{\epsilon \in R| \epsilon^2 = 0\} for the standard infinitesimal interval. For X𝒯X \in \mathcal{T} any object (any space in 𝒯\mathcal{T}), the tangent bundle of XX is the morphism

p:TXX p : T X \to X

with

  • TX∶−X DT X \coloneq X^D the internal hom of DD into XX;

  • p=ev 0p = ev_0 the evaluation map at the origin of DD

    ev 0:(UvX D)(U×*Id×0U×Dv¯X)ev_0 : (U \stackrel{v}{\to} X^D) \mapsto (U \times {*} \stackrel{Id \times 0}{\to} U \times D \stackrel{\bar v}{\to} X),

    where v¯\bar v is the hom-adjunct of vv.

This definition captures elegantly and usefully the notion of tangent vectors as infinitesimal curves. But it is not guaranteed that the fibers of a synthetic tangent bundle X DX^D are fiberwise linear, i.e. are fiberwise RR-modules the way one expects. Objects XX for which this is true are microlinear spaces in 𝒯\mathcal{T}. See there for more details.

A smooth topos 𝒯\mathcal{T} is called a well-adapted model for synthetic differential geometry if there is a full and faithful embedding Diff 𝒯\hookrightarrow \mathcal{T} of the cageory of manifolds into 𝒯\mathcal{T}.

Typically, for well adapted models, under this embedding

  • manifolds are microlinear spaces

  • the synthetic definition of tangent bundle X DX^D for XX a manifold does coincide with the ordinary notion of TXT X.

Let 𝕃=(C Ring fin) op\mathbb{L} = (C^\infty Ring^{fin})^{op} be the category of smooth loci. For MM a manifold, the exponential M DM^D does exist in 𝕃\mathbb{L} and is isomorphic to the ordinary tangent bundle TXT X of XX. (For instance MSIA, chapter II, prop 1.12.

There are well-adapted smooth toposes 𝒵\mathcal{Z} and \mathcal{B} presented as categories of sheaves on 𝕃\mathbb{L}: the first for the Grothendieck topology where covers are finite open covers, the second where covers are finite open covers and projections (MSIA, chapter VI). Both topologies are subcanonical, hence the Yoneda embedding Y:𝕃Sh(𝕃)Y : \mathbb{L} \to Sh(\mathbb{L}) does preserve the above property.

Hence in these models for XDiffX \in Diff a manifold, TXDiffT X \in Diff its ordinary tangent bundle and s:DiffSh(𝕃)s : Diff \to Sh(\mathbb{L}) the full and faithful embedding, we have isomorphisms

(s(X)) Ds(TX) (s(X))^D \simeq s(T X)

which respect the bundle maps.

In supergeometry

The tangent bundle of a manifold XX may be interpreted as a supermanifold in which XX has degree 00 and the tangent vectors have degree 11. See shifted tangent bundle.

For other generalized smooth spaces

There are useful categories of generalized smooth spaces which are neither categories of manifolds nor smooth toposes modeling synthetic differential geometry. But most of them admit useful notions of tangent bundles, too, sometimes more than one.

See Frölicher space and diffeological space for the definitions in their context.

Examples of sequences of local structures

geometrypointfirst order infinitesimal\subsetformal = arbitrary order infinitesimal\subsetlocal = stalkwise\subsetfinite
\leftarrow differentiationintegration \to
smooth functionsderivativeTaylor seriesgermsmooth function
curve (path)tangent vectorjetgerm of curvecurve
smooth spaceinfinitesimal neighbourhoodformal neighbourhoodgerm of a spaceopen neighbourhood
function algebrasquare-0 ring extensionnilpotent ring extension/formal completionring extension
arithmetic geometry𝔽 p\mathbb{F}_p finite field p\mathbb{Z}_p p-adic integers (p)\mathbb{Z}_{(p)} localization at (p)\mathbb{Z} integers
Lie theoryLie algebraformal grouplocal Lie groupLie group
symplectic geometryPoisson manifoldformal deformation quantizationlocal strict deformation quantizationstrict deformation quantization

References

A textbook account of tangent bundles in the context of synthetic differential geometry is in

Further discussion of axiomatizations in this context is in

Discussion for diffeological spaces is in

Revised on June 22, 2017 23:58:18 by Elves? (172.222.163.170)