nLab tangent bundle

This entry is about tangent vectors on differentiable manifolds and the bundle they form. For tangents and the tangent function see there.

Idea

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

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

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 $X$ at $x \in X$ is an element of $T_x X$.

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

• A tangent vector field on $X$ is a section of $T X$.

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 $X$ be a differentiable manifold of dimension $n$ and let $x \in X$ be a point. On the set of smooth functions of the form

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

such that

$$\gamma(0) = x$$

define the relations

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

and

$$(\gamma_1 \sim' \gamma_2) \coloneqq \underset{ { { \mathbb{R}^n \underoverset{}{\phi \, \text{chart}}{\to} U_i \subset X } } \atop { U_i \supset \{x\} } }{ \forall } \left( \frac{d}{d t}(\phi^{-1}\circ \gamma_1)(0) = \frac{d}{d t}(\phi^{-1}\circ \gamma_2)(0) \right)$$

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

Lemma

(tangency is equivalence relation)

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

Proof

First to see that they coincide, we need to show that if the derivatives in question coincide in one chart $\mathbb{R}^n \underoverset{\simeq}{\phi_i}{\to} U_i \subset X$, that then they coincide also in any other chart $\mathbb{R}^n \underoverset{\simeq}{\phi_j}{\to} U_j \subset X$.

For brevity, write

$$U_{i j} \coloneqq U_i \cap U_j$$

for the intersection of the two charts.

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

$$V \coloneqq \gamma_1^{-1}( U_{i j} ) \cap \gamma_2^{-1}(U_{i j}) \;\subset\; \mathbb{R}$$

of $0 \in \mathbb{R}$ and consider the derivatives of the functions

$$\gamma_n^i \;\coloneqq\; (\phi_i^{-1}\vert_{U_{i j}} \circ \gamma_n\vert_{V}) \;\colon\; V \longrightarrow \phi_i^{-1}(U_{i j}) \subset \mathbb{R}^n$$

and

$$\gamma_n^j \;\coloneqq\; (\phi_j^{-1}\vert_{U_{i j}} \circ \gamma_n \vert _{V}) \;\colon\; V \longrightarrow \phi_j^{-1}(U_{i j}) \subset \mathbb{R}^n \,.$$

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

$$\alpha \;\coloneqq\; \phi_i^{-1}(U_{i j}) \underoverset{\simeq}{\phi_i}{\longrightarrow} U_{i j} \underoverset{\simeq}{\phi_j^{-1}}{\longrightarrow} \phi_j^{-1}(U_{i j})$$

such that

$$\gamma_n^j = \alpha \circ \gamma_n^i$$

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

$$\frac{d}{d t} \gamma_n^j \;=\; (D \alpha) \circ \left(\frac{d}{d t} \gamma_n^i \right) \,.$$

Since $\alpha$ is a diffeomorphism and since derivatives of diffeomorphisms are linear isomorphisms, this says that the derivative of $\gamma_n^j$ is related to that of $\gamma_n^i$ by a linear isomorphism , and hence

$$\left( \frac{d}{d t} \gamma_1^i = \frac{d}{d t} \gamma_2^i \right) \;\Leftrightarrow\; \left( \frac{d}{d t} \gamma_1^j = \frac{d}{d t} \gamma_2^j \right) \,.$$

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

Definition

(tangent vector)

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

The set of all tangent vectors at $x \in X$ is denoted $T_x X$.

Lemma

(real vector space structure on tangent vectors)

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

Then there is induced a bijection of sets

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

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

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

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

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

of the gluing function of the two charts at the point $x$:

$$\array{ \mathbb{R}^n && \overset{ d \left((\phi')^{-1} \circ \phi\right)_{\phi^{-1}(x)}}{\longrightarrow} && \mathbb{R}^n \\ & {}_{\mathllap{\simeq}}\searrow && \swarrow_{\mathrlap{\simeq}} \\ && T_x X } \,.$$

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

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

$$\left\{ g_{i j} \coloneqq d( \phi_j^{-1} \circ \phi_i )_{\phi_i^{-1}(-)} \colon U_i \cap U_j \longrightarrow GL(n,\mathbb{R}) \right\}_{i,j \in I}$$

defined this way satisfy the following Cech cocycle conditions for all $i,j \in I$, $x \in U_i \cap U_j$

1. $g_{i i}(x) = id_{\mathbb{R}^n}$;

2. $g_{j k}\circ g_{i j}(x) = g_{i k}(x)$.

Proof

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

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

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

Similarly the Cech cocycle condition follows by the chain rule:

$$\begin{aligned} g_{j k} \circ g_{i j}(x) & = d( \phi_k^{-1} \circ \phi_j )_{\phi_j^{-1}(x)} \circ d( \phi_j^{-1} \circ \phi_i )_{\phi_i^{-1}(x)} \\ & = d( \phi_k^{-1} \circ \phi_j \circ \phi_j^{-1} \circ \phi_i )_{\phi_i^{-1}(x)} \\ & = d( \phi_k^{-1} \circ \phi_i )_{\phi_i^{-1}(x)} \\ &= g_{i k}(x) \end{aligned} \,.$$

Definition

(tangent space)

For $X$ a differentiable manifold and $x \in X$ a point, then the tangent space of $X$ at $x$ is the set $T_x X$ of tangent vectors at $x$ (def. ) regarded as a real vector space via lemma .

Example

(tangent bundle of Euclidean space)

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

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

$$\gamma' \colon t \mapsto \gamma(t) + (x-y) \,.$$

This gives a linear bijection

$$\phi_{x,y} \colon T_x X \overset{\simeq}{\longrightarrow} T_y X$$

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

$$\phi_{y,z} \circ \phi_{x,y} = \phi_{x,z} \;\colon\; T_x X \longrightarrow T_y Y \,.$$

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

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

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

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

equipped with the projection on the first factor

$$\array{ T \mathbb{R}^n = \mathbb{R}^n \times \mathbb{R}^n \\ \downarrow^{\mathrlap{\pi = pr_1}} \\ \mathbb{R}^n } \,,$$

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

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

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

This is called the tangent bundle of the Euclidean space $\mathbb{R}^n$ 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 $\mathbb{R}^n$ to its tangent bundle $T \mathbb{R}^n$ according to def. ;

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

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

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

$$d (f \circ \gamma)_t = (d f_{\gamma(x)}) \circ d\gamma$$

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

$$\array{ \mathbb{R}^{n_1} \times \mathbb{R}^{n_1} &\overset{ d f }{\longrightarrow}& \mathbb{R}^{n_2} \times \mathbb{R}^{n_2} \\ (x,\vec v) &\mapsto& (f(x), d f_x(\vec v)) } \,,$$

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

$$\array{ && Y \\ & {}^{\mathllap{f}}\nearrow && \searrow^{\mathrlap{g}} \\ X && \underset{ g\circ f}{\longrightarrow} && Z } \phantom{AAA} \mapsto \phantom{AAA} \array{ && T Y \\ & {}^{\mathllap{d f}}\nearrow && \searrow^{\mathrlap{d g}} \\ T X && \underset{d(g \circ f)}{\longrightarrow} && T Z } \,.$$

In the language of category theory this says that the assignment

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

is an endofunctor on the category CartSp whose

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

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

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 $X$ be a differentiable manifold with atlas $\left\{ \mathbb{R}^n \underoverset{\simeq}{\phi_i}{\to} U_i \subset X\right\}_{i \in I}$.

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

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

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

$$\array{ \mathbb{R}^{2n} \simeq \mathbb{R}^n \times \mathbb{R}^n & \overset{d \phi}{\longrightarrow} & T X \\ (x, \vec v) &\overset{\phantom{AAA}}{\mapsto}& [ t \mapsto \phi(\phi^{-1}(x) + t \vec v) ] }$$

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

Equipped with the function

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

this is called the tangent bundle of $X$.

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

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

(Notice that, by examples , each $U_i \times \mathbb{R}^n \simeq T U_i$ is the tangent bundle of the chart $U_i \simeq \mathbb{R}^n$.)

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

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

Lemma

(tangent bundle is differentiable vector bundle)

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

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

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

Moreover, the projection $\pi \colon T X \to X$ is a $p$-times continuously differentiable function.

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

Proof

First to see that $T X$ is Hausdorff:

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

Now to see that $T X$ is paracompact.

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

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

$$\left\{ \pi(U_i) \cap V_j \subset X \right\}_{i \in I, j \in J} \,.$$

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

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

Notice that for each $j' \in J'$ the product topological space $V'_{j'} \times \mathbb{R}^n \subset \mathbb{R}^{2n}$ 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

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

has a locally finite refinement

$$\{W_{k_{j'}} \subset V'_{j'} \times \mathbb{R}^n \}_{k_{j'} \in K_{j'}} \,.$$

We claim now that

$$\{ W_{k_{j'}} \subset T X \}_{j' \in J', k_{j'} \in K_{j'}}$$

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

This shows that $T X$ is paracompact.

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

Proposition

(differentials of differentiable functions between differentiable manifolds)

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

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

descends at each point $x \in X$ to the tangency equivalence relation (def. , lemma ) to yield a function on sets of tangent vectors (def. ), called the differential $d f_x$ of $f$ at $x$

$$d f|_{x} \;\colon\; T_x X \longrightarrow T_{f(x)} Y \,.$$

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 , def. ;

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

   $$d f \;\colon\; T X \longrightarrow T Y \,.$$

3. (chain rule) The assignment $f \mapsto d f$ respects composition in that for $X$, $Y$, $Z$ three differentiable manifolds and for

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

   two composable differentiable functions then their differentials satisfy

   $$d(g \circ f) = (d g) \circ (d f) \,.$$

Proof

All statements are to be tested on charts of an atlas for $X$ and for $Y$. On these charts the statement reduces to that of example .

Remark

(tangent functor)

In the language of category theory the statement of prop. says that forming tangent bundles $T X$ of differentiable manifolds $X$ and differentials $d f$ of differentiable functions $f \colon X \to Y$ constitutes a functor

$$T \;\colon\; Diff \longrightarrow Vect(Diff)$$

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

Definition

(vector field)

Let $X$ be a differentiable manifold with differentiable tangent bundle $T X \to X$ (def. ).

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

Remark

(notation for tangent vectors in a chart)

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

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

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

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

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 $f,g$ on a 2-dimensional manifold are used as coordinate functions for a local chart (i.e. so that $x^1 = f$ and $x^2 = g$ ), then one writes

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

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

Via derivations of smooth functions

Remark

(derivations of smooth functions are vector fields)

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

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

$$\partial_v \;\colon\; C^\infty(X) \longrightarrow C^\infty(X)$$

by

$$\partial f \;\colon\; x \mapsto \frac{d}{d t} (f \circ \gamma_{v_x})_0$$

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

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

$$\array{ \Gamma_X(T X) &\longrightarrow& Der(C^\infty(X)) \\ v &\mapsto& \partial_v } \,.$$

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

For manifolds of the class $C^k$, $0\lt k\lt \infty$, the definition of a tangent vector as a derivation of the algebra of functions (derivations of smooth functions are vector fields) remains valid if one strengthens the definition of a derivation $D$ by adjoining, on top of the algebraic condition of the product rule

$$D (f \cdot g) = (D f) g + f (D g)$$

also the algebraic incarnation of the chain rule:

$$D(f(g_1,\ldots,g_m)) \;=\; \sum_i (\partial_i f)(g_1,\ldots,g_m)D(g_i).$$

See Newns-Walker 56.

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

$$p : T X \to X$$

with

• $T X \coloneq X^D$ the internal hom of $D$ into $X$;

• $p = ev_0$ the evaluation map at the origin of $D$

  $ev_0 : (U \stackrel{v}{\to} X^D) \mapsto (U \times {*} \stackrel{Id \times 0}{\to} U \times D \stackrel{\bar v}{\to} X)$,

  where $\bar v$ is the hom-adjunct of $v$.

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 $X^D$ are fiberwise linear, i.e. are fiberwise $R$-modules the way one expects. Objects $X$ for which this is true are microlinear spaces in $\mathcal{T}$. See there for more details.

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

Typically, for well adapted models, under this embedding

• manifolds are microlinear spaces

• the synthetic definition of tangent bundle $X^D$ for $X$ a manifold does coincide with the ordinary notion of $T X$.

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

There are well-adapted smooth topoi $\mathcal{Z}$ and $\mathcal{B}$ presented as categories of sheaves on $\mathbb{L}$: 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 $Y : \mathbb{L} \to Sh(\mathbb{L})$ does preserve the above property.

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

$$(s(X))^D \simeq s(T X)$$

which respect the bundle maps.

In supergeometry

The tangent bundle of a manifold $X$ may be interpreted as a supermanifold in which $X$ has degree $0$ and the tangent vectors have degree $1$. 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 topoi 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.

Related concepts

• tangent

• synthetic tangent bundle, kinematic tangent bundle, operational tangent bundle

• cotangent bundle

• normal bundle

• Euler vector field

• G-structure

• stable tangent bundle

• tangent cone

• Lie algebra

Examples of sequences of local structures

geometry	point	first order infinitesimal	$\subset$	formal = arbitrary order infinitesimal	$\subset$	local = stalkwise	$\subset$	finite
			$\leftarrow$ differentiation		integration $\to$
smooth functions		derivative		Taylor series		germ		smooth function
curve (path)		tangent vector		jet		germ of curve		curve
smooth space		infinitesimal neighbourhood		formal neighbourhood		germ of a space		open neighbourhood
function algebra		square-0 ring extension		nilpotent ring extension/formal completion				ring extension
arithmetic geometry	$\mathbb{F}_p$ finite field			$\mathbb{Z}_p$ p-adic integers		$\mathbb{Z}_{(p)}$ localization at (p)		$\mathbb{Z}$ integers
Lie theory		Lie algebra		formal group		local Lie group		Lie group
symplectic geometry		Poisson manifold		formal deformation quantization		local strict deformation quantization		strict deformation quantization

• in homotopy theory/Goodwillie calculus: tangent (∞,1)-category

References

An early account of tangent vectors as derivations, including the $C^k$-case for $0\lt k\lt \infty$ is in

• W. F. Newns, A. G. Walker, Tangent Planes To a Differentiable Manifold. Journal of the London Mathematical Society s1-31:4 (1956), 400–407 (doi:10.1112/jlms/s1-31.4.400)

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

• Ieke Moerdijk, Gonzalo E. Reyes, Models for Smooth Infinitesimal Analysis Springer (1991)

Further discussion of axiomatizations in this context is in

• J.R.B. Cockett, Geoff Cruttwell, Differential structure, tangent structure, and SDG (2012) (pdf)

Discussion for diffeological spaces is in

• Carlos Torre, A tangent bundle for diffeological spaces (arXiv:math/9801046)