# Tangent bundle

A tangent bundle is the collection of all of the tangent spaces for all points on a manifold, structured in a way that it forms a new manifold itself. Formally, in differential geometry, the tangent bundle of a differentiable manifold $M$ is a manifold $TM$ which assembles all the tangent vectors in $M$. As a set, it is given by the disjoint union of the tangent spaces of $M$. That is,
    ${\begin{aligned}TM&=\bigsqcup _{x\in M}T_{x}M\\&=\bigcup _{x\in M}\left\{x\right\}\times T_{x}M\\&=\bigcup _{x\in M}\left\{(x,y)\mid y\in T_{x}M\right\}\\&=\left\{(x,y)\mid x\in M,\,y\in T_{x}M\right\}\end{aligned}}$
where $T_{x}M$ denotes the tangent space to $M$ at the point $x$. So, an element of $TM$ can be thought of as a pair $(x,v)$, where $x$ is a point in $M$ and $v$ is a tangent vector to $M$ at $x$.
There is a natural projection
    $\pi :TM\twoheadrightarrow M$
defined by $\pi (x,v)=x$. This projection maps each element of the tangent space $T_{x}M$ to the single point $x$.
The tangent bundle comes equipped with a natural topology (described in a section below). With this topology, the tangent bundle to a manifold is the prototypical example of a vector bundle (which is a fiber bundle whose fibers are vector spaces). A section of $TM$ is a vector field on $M$, and the dual bundle to $TM$ is the cotangent bundle, which is the disjoint union of the cotangent spaces of $M$. By definition, a manifold $M$ is parallelizable if and only if the tangent bundle is trivial. By definition, a manifold $M$ is framed if and only if the tangent bundle $TM$ is stably trivial, meaning that for some trivial bundle $E$ the Whitney sum $TM\oplus E$ is trivial. For example, the n-dimensional sphere S$^n$ is framed for all n, but parallelizable only for n = 1, 3, 7 (by results of Bott-Milnor and Kervaire).

## Role

One of the main roles of the tangent bundle is to provide a domain and range for the derivative of a smooth function. Namely, if $f:M\rightarrow N$ is a smooth function, with $M$ and $N$ smooth manifolds, its derivative is a smooth function $Df:TM\rightarrow TN$.

## Topology and smooth structure

The tangent bundle comes equipped with a natural topology (not the disjoint union topology) and smooth structure so as to make it into a manifold in its own right. The dimension of $TM$ is twice the dimension of $M$.
Each tangent space of an n-dimensional manifold is an n-dimensional vector space. If $U$ is an open contractible subset of $M$, then there is a diffeomorphism $TU\to U\times \mathbb {R} ^{n}$ which restricts to a linear isomorphism from each tangent space $T_{x}U$ to $\{x\}\times \mathbb {R} ^{n}$. As a manifold, however, $TM$ is not always diffeomorphic to the product manifold $M\times \mathbb {R} ^{n}$. When it is of the form $M\times \mathbb {R} ^{n}$, then the tangent bundle is said to be trivial. Trivial tangent bundles usually occur for manifolds equipped with a 'compatible group structure'; for instance, in the case where the manifold is a Lie group. The tangent bundle of the unit circle is trivial because it is a Lie group (under multiplication and its natural differential structure). It is not true however that all spaces with trivial tangent bundles are Lie groups; manifolds which have a trivial tangent bundle are called parallelizable. Just as manifolds are locally modeled on Euclidean space, tangent bundles are locally modeled on $U\times \mathbb {R} ^{n}$, where $U$ is an open subset of Euclidean space.
If M is a smooth n-dimensional manifold, then it comes equipped with an atlas of charts $(U_{\alpha },\phi _{\alpha })$, where $U_{\alpha }$ is an open set in $M$ and
    $\phi _{\alpha }:U_{\alpha }\to \mathbb {R} ^{n}$
is a diffeomorphism. These local coordinates on $U_{\alpha }$ give rise to an isomorphism $T_{x}M\rightarrow \mathbb {R} ^{n}$ for all $x\in U_{\alpha }$. We may then define a map
    ${\widetilde {\phi }}_{\alpha }:\pi ^{-1}\left(U_{\alpha }\right)\to \mathbb {R} ^{2n}$
by
    ${\widetilde {\phi }}_{\alpha }\left(x,v^{i}\partial _{i}\right)=\left(\phi _{\alpha }(x),v^{1},\cdots ,v^{n}\right)$
We use these maps to define the topology and smooth structure on $TM$. A subset $A$ of $TM$ is open if and only if
    ${\widetilde {\phi }}_{\alpha }\left(A\cap \pi ^{-1}\left(U_{\alpha }\right)\right)$
is open in $\mathbb {R} ^{2n}$ for each $\alpha .$ These maps are homeomorphisms between open subsets of $TM$ and $\mathbb {R} ^{2n}$ and therefore serve as charts for the smooth structure on $TM$. The transition functions on chart overlaps $\pi ^{-1}\left(U_{\alpha }\cap U_{\beta }\right)$ are induced by the Jacobian matrices of the associated coordinate transformation and are therefore smooth maps between open subsets of $\mathbb {R} ^{2n}$.
The tangent bundle is an example of a more general construction called a vector bundle (which is itself a specific kind of fiber bundle). Explicitly, the tangent bundle to an $n$-dimensional manifold $M$ may be defined as a rank $n$ vector bundle over $M$ whose transition functions are given by the Jacobian of the associated coordinate transformations.

## Examples

The simplest example is that of $\mathbb {R} ^{n}$. In this case the tangent bundle is trivial: each $T_{x}\mathbf {\mathbb {R} } ^{n}$ is canonically isomorphic to $T_{0}\mathbb {R} ^{n}$ via the map $\mathbb {R} ^{n}\to \mathbb {R} ^{n}$ which subtracts $x$, giving a diffeomorphism $T\mathbb {R} ^{n}\to \mathbb {R} ^{n}\times \mathbb {R} ^{n}$.
Another simple example is the unit circle, $S^{1}$ (see picture above). The tangent bundle of the circle is also trivial and isomorphic to $S^{1}\times \mathbb {R}$. Geometrically, this is a cylinder of infinite height.
The only tangent bundles that can be readily visualized are those of the real line $\mathbb {R}$ and the unit circle $S^{1}$, both of which are trivial. For 2-dimensional manifolds the tangent bundle is 4-dimensional and hence difficult to visualize.
A simple example of a nontrivial tangent bundle is that of the unit sphere $S^{2}$: this tangent bundle is nontrivial as a consequence of the hairy ball theorem. Therefore, the sphere is not parallelizable.

## Vector fields

A smooth assignment of a tangent vector to each point of a manifold is called a vector field. Specifically, a vector field on a manifold $M$ is a smooth map
    $V\colon M\to TM$
such that $V(x)=(x,V_{x})$ with $V_{x}\in T_{x}M$ for every $x\in M$. In the language of fiber bundles, such a map is called a section. A vector field on $M$ is therefore a section of the tangent bundle of $M$.
The set of all vector fields on $M$ is denoted by $\Gamma (TM)$. Vector fields can be added together pointwise
    $(V+W)_{x}=V_{x}+W_{x}$
and multiplied by smooth functions on M
    $(fV)_{x}=f(x)V_{x}$
to get other vector fields. The set of all vector fields $\Gamma (TM)$ then takes on the structure of a module over the commutative algebra of smooth functions on M, denoted $C^{\infty }(M)$.
A local vector field on $M$ is a local section of the tangent bundle. That is, a local vector field is defined only on some open set $U\subset M$ and assigns to each point of $U$ a vector in the associated tangent space. The set of local vector fields on $M$ forms a structure known as a sheaf of real vector spaces on $M$.
The above construction applies equally well to the cotangent bundle – the differential 1-forms on $M$ are precisely the sections of the cotangent bundle $\omega \in \Gamma (T^{*}M)$, $\omega :M\to T^{*}M$ that associate to each point $x\in M$ a 1-covector $\omega _{x}\in T_{x}^{*}M$, which map tangent vectors to real numbers: $\omega _{x}:T_{x}M\to \mathbb {R}$. Equivalently, a differential 1-form $\omega \in \Gamma (T^{*}M)$ maps a smooth vector field $X\in \Gamma (TM)$ to a smooth function $\omega (X)\in C^{\infty }(M)$.

## Higher-order tangent bundles

Since the tangent bundle $TM$ is itself a smooth manifold, the second-order tangent bundle can be defined via repeated application of the tangent bundle construction:
    $T^{2}M=T(TM).\,$
In general, the $k$th order tangent bundle $T^{k}M$ can be defined recursively as $T\left(T^{k-1}M\right)$.
A smooth map $f:M\rightarrow N$ has an induced derivative, for which the tangent bundle is the appropriate domain and range $Df:TM\rightarrow TN$. Similarly, higher-order tangent bundles provide the domain and range for higher-order derivatives $D^{k}f:T^{k}M\to T^{k}N$.
A distinct but related construction are the jet bundles on a manifold, which are bundles consisting of jets.

## Canonical vector field on tangent bundle

On every tangent bundle $TM$, considered as a manifold itself, one can define a canonical vector field $V:TM\rightarrow T^{2}M$ as the diagonal map on the tangent space at each point. This is possible because the tangent space of a vector space W is naturally a product, $TW\cong W\times W,$ since the vector space itself is flat, and thus has a natural diagonal map $W\to TW$ given by $w\mapsto (w,w)$ under this product structure. Applying this product structure to the tangent space at each point and globalizing yields the canonical vector field. Informally, although the manifold $M$ is curved, each tangent space at a point $x$, $T_{x}M\approx \mathbb {R} ^{n}$, is flat, so the tangent bundle manifold $TM$ is locally a product of a curved $M$ and a flat $\mathbb {R} ^{n}.$ Thus the tangent bundle of the tangent bundle is locally (using $\approx$ for "choice of coordinates" and $\cong$ for "natural identification"):
    $T(TM)\approx T(M\times \mathbb {R} ^{n})\cong TM\times T(\mathbb {R} ^{n})\cong TM\times (\mathbb {R} ^{n}\times \mathbb {R} ^{n})$
and the map $TTM\to TM$ is the projection onto the first coordinates:
    $(TM\to M)\times (\mathbb {R} ^{n}\times \mathbb {R} ^{n}\to \mathbb {R} ^{n}).$
Splitting the first map via the zero section and the second map by the diagonal yields the canonical vector field.
If $(x,v)$ are local coordinates for $TM$, the vector field has the expression
    $V=\sum _{i}\left.v^{i}{\frac {\partial }{\partial v^{i}}}\right|_{(x,v)}.$
More concisely, $(x,v)\mapsto (x,v,0,v)$ – the first pair of coordinates do not change because it is the section of a bundle and these are just the point in the base space: the last pair of coordinates are the section itself. This expression for the vector field depends only on $v$, not on $x$, as only the tangent directions can be naturally identified.
Alternatively, consider the scalar multiplication function:
    ${\begin{cases}\mathbb {R} \times TM\to TM\\(t,v)\longmapsto tv\end{cases}}$
The derivative of this function with respect to the variable $\mathbb {R}$ at time $t=1$ is a function $V:TM\rightarrow T^{2}M$, which is an alternative description of the canonical vector field.
The existence of such a vector field on $TM$ is analogous to the canonical one-form on the cotangent bundle. Sometimes $V$ is also called the Liouville vector field, or radial vector field. Using $V$ one can characterize the tangent bundle. Essentially, $V$ can be characterized using 4 axioms, and if a manifold has a vector field satisfying these axioms, then the manifold is a tangent bundle and the vector field is the canonical vector field on it. See for example, De León et al.

## Lifts

There are various ways to lift objects on $M$ into objects on $TM$. For example, if $\gamma$ is a curve in $M$, then $\gamma '$ (the tangent of $\gamma$) is a curve in $TM$. In contrast, without further assumptions on $M$ (say, a Riemannian metric), there is no similar lift into the cotangent bundle.
The vertical lift of a function $f:M\rightarrow \mathbb {R}$ is the function $f^{\vee }:TM\rightarrow \mathbb {R}$ defined by $f^{\vee }=f\circ \pi$, where $\pi :TM\rightarrow M$ is the canonical projection.