Tangent bundle

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 ${\displaystyle TM}$ is twice the dimension of ${\displaystyle M}$.

Each tangent space of an n-dimensional manifold is an n-dimensional vector space. If ${\displaystyle U}$ is an open contractible subset of ${\displaystyle M}$, then there is a diffeomorphism ${\displaystyle TU\to U\times {\mathbb{R}}^n}$ which restricts to a linear isomorphism from each tangent space ${\displaystyle T_{x}U}$ to ${\displaystyle \{x\}\times {\mathbb{R}}^n}$. As a manifold, however, ${\displaystyle TM}$ is not always diffeomorphic to the product manifold ${\displaystyle M\times {\mathbb{R}}^n}$. When it is of the form ${\displaystyle M\times {\mathbb{R}}^n}$, 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 ${\displaystyle U\times {\mathbb{R}}^n}$, where ${\displaystyle U}$ is an open subset of Euclidean space.

If M is a smooth n-dimensional manifold, then it comes equipped with an atlas of charts ${\displaystyle (U_{\alpha },{\varphi }_{\alpha })}$, where ${\displaystyle U_{\alpha }}$ is an open set in ${\displaystyle M}$ and

${\displaystyle {\varphi }_{\alpha }:U_{\alpha }\to {\mathbb{R}}^n}$

is a diffeomorphism. These local coordinates on ${\displaystyle U_{\alpha }}$ give rise to an isomorphism ${\displaystyle T_{x}M\to {\mathbb{R}}^n}$ for all ${\displaystyle x\in U_{\alpha }}$. We may then define a map

${\displaystyle {\tilde{\varphi }}_{\alpha }:{\pi }^{-1}\left(U_{\alpha }\right)\to {\mathbb{R}}^{2n}}$

by

${\displaystyle {\tilde{\varphi }}_{\alpha }\left(x,v^i{\mathrm{\partial }}_i\right)=\left({\varphi }_{\alpha }(x),v^1,\cdots ,v^n\right)}$

We use these maps to define the topology and smooth structure on ${\displaystyle TM}$. A subset ${\displaystyle A}$ of ${\displaystyle TM}$ is open if and only if

${\displaystyle {\tilde{\varphi }}_{\alpha }\left(A\cap {\pi }^{-1}\left(U_{\alpha }\right)\right)}$

is open in ${\displaystyle {\mathbb{R}}^{2n}}$ for each ${\displaystyle \alpha .}$ These maps are homeomorphisms between open subsets of ${\displaystyle TM}$ and ${\displaystyle {\mathbb{R}}^{2n}}$ and therefore serve as charts for the smooth structure on ${\displaystyle TM}$. The transition functions on chart overlaps ${\displaystyle {\pi }^{-1}\left(U_{\alpha }\cap U_{\beta }\right)}$ are induced by the Jacobian matrices of the associated coordinate transformation and are therefore smooth maps between open subsets of ${\displaystyle {\mathbb{R}}^{2n}}$.

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 ${\displaystyle n}$-dimensional manifold ${\displaystyle M}$ may be defined as a rank ${\displaystyle n}$ vector bundle over ${\displaystyle M}$ whose transition functions are given by the Jacobian of the associated coordinate transformations.

Examples

The simplest example is that of ${\displaystyle {\mathbb{R}}^n}$. In this case the tangent bundle is trivial: each ${\displaystyle T_x{\mathbb{R}}^n}$ is canonically isomorphic to ${\displaystyle T_{\mathbf{0}}{\mathbb{R}}^n}$ via the map ${\displaystyle {\mathbb{R}}^n\to {\mathbb{R}}^n}$ which subtracts ${\displaystyle x}$, giving a diffeomorphism ${\displaystyle T{\mathbb{R}}^n\to {\mathbb{R}}^n\times {\mathbb{R}}^n}$.

Another simple example is the unit circle, ${\displaystyle S^1}$ (see picture above). The tangent bundle of the circle is also trivial and isomorphic to ${\displaystyle S^1\times \mathbb{R}}$. Geometrically, this is a cylinder of infinite height.

The only tangent bundles that can be readily visualized are those of the real line ${\displaystyle \mathbb{R}}$ and the unit circle ${\displaystyle S^1}$, 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 ${\displaystyle S^2}$: 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 ${\displaystyle M}$ is a smooth map

${\displaystyle V:M\to TM}$

such that ${\displaystyle V(x)=(x,V_x)}$ with ${\displaystyle V_x\in T_{x}M}$ for every ${\displaystyle x\in M}$. In the language of fiber bundles, such a map is called a section. A vector field on ${\displaystyle M}$ is therefore a section of the tangent bundle of ${\displaystyle M}$.

The set of all vector fields on ${\displaystyle M}$ is denoted by ${\displaystyle \mathrm{\Gamma }(TM)}$. Vector fields can be added together pointwise

${\displaystyle (V+W{)}_x=V_x+W_x}$

and multiplied by smooth functions on M

${\displaystyle (fV{)}_x=f(x)V_x}$

to get other vector fields. The set of all vector fields ${\displaystyle \mathrm{\Gamma }(TM)}$ then takes on the structure of a module over the commutative algebra of smooth functions on M, denoted ${\displaystyle C^{\mathrm{\infty }}(M)}$.

A local vector field on ${\displaystyle M}$ is a local section of the tangent bundle. That is, a local vector field is defined only on some open set ${\displaystyle U\subset M}$ and assigns to each point of ${\displaystyle U}$ a vector in the associated tangent space. The set of local vector fields on ${\displaystyle M}$ forms a structure known as a sheaf of real vector spaces on ${\displaystyle M}$.

The above construction applies equally well to the cotangent bundle – the differential 1-forms on ${\displaystyle M}$ are precisely the sections of the cotangent bundle ${\displaystyle \omega \in \mathrm{\Gamma }(T^{\ast }M)}$, ${\displaystyle \omega :M\to T^{\ast }M}$ that associate to each point ${\displaystyle x\in M}$ a 1-covector ${\displaystyle {\omega }_x\in T_x^{\ast }M}$, which map tangent vectors to real numbers: ${\displaystyle {\omega }_x:T_{x}M\to \mathbb{R}}$. Equivalently, a differential 1-form ${\displaystyle \omega \in \mathrm{\Gamma }(T^{\ast }M)}$ maps a smooth vector field ${\displaystyle X\in \mathrm{\Gamma }(TM)}$ to a smooth function ${\displaystyle \omega (X)\in C^{\mathrm{\infty }}(M)}$.

Higher-order tangent bundles

Since the tangent bundle ${\displaystyle TM}$ is itself a smooth manifold, the second-order tangent bundle can be defined via repeated application of the tangent bundle construction:

${\displaystyle T^{2}M=T(TM).}$

In general, the ${\displaystyle k}$th order tangent bundle ${\displaystyle T^{k}M}$ can be defined recursively as ${\displaystyle T\left(T^{k-1}M\right)}$.

A smooth map ${\displaystyle f:M\to N}$ has an induced derivative, for which the tangent bundle is the appropriate domain and range ${\displaystyle Df:TM\to TN}$. Similarly, higher-order tangent bundles provide the domain and range for higher-order derivatives ${\displaystyle D^{k}f:T^{k}M\to T^{k}N}$.

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 ${\displaystyle TM}$, considered as a manifold itself, one can define a canonical vector field ${\displaystyle V:TM\to T^{2}M}$ 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, ${\displaystyle TW\cong W\times W,}$ since the vector space itself is flat, and thus has a natural diagonal map ${\displaystyle W\to TW}$ given by ${\displaystyle w\mapsto (w,w)}$ 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 ${\displaystyle M}$ is curved, each tangent space at a point ${\displaystyle x}$, ${\displaystyle T_{x}M\approx {\mathbb{R}}^n}$, is flat, so the tangent bundle manifold ${\displaystyle TM}$ is locally a product of a curved ${\displaystyle M}$ and a flat ${\displaystyle {\mathbb{R}}^n.}$ Thus the tangent bundle of the tangent bundle is locally (using ${\displaystyle \approx }$ for "choice of coordinates" and ${\displaystyle \cong }$ for "natural identification"):

${\displaystyle T(TM)\approx T(M\times {\mathbb{R}}^n)\cong TM\times T({\mathbb{R}}^n)\cong TM\times ({\mathbb{R}}^n\times {\mathbb{R}}^n)}$

and the map ${\displaystyle TTM\to TM}$ is the projection onto the first coordinates:

${\displaystyle (TM\to M)\times ({\mathbb{R}}^n\times {\mathbb{R}}^n\to {\mathbb{R}}^n).}$

Splitting the first map via the zero section and the second map by the diagonal yields the canonical vector field.

If ${\displaystyle (x,v)}$ are local coordinates for ${\displaystyle TM}$, the vector field has the expression

${\displaystyle V=\sum _i{v^i\frac{\mathrm{\partial }}{\mathrm{\partial }{v}^i}|}_{(x,v)}.}$

More concisely, ${\displaystyle (x,v)\mapsto (x,v,0,v)}$ – 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 ${\displaystyle v}$, not on ${\displaystyle x}$, as only the tangent directions can be naturally identified.

Alternatively, consider the scalar multiplication function:

${\displaystyle \{\begin{array}{l}\mathbb{R}\times TM\to TM\\ (t,v)⟼tv\end{array}}$

The derivative of this function with respect to the variable ${\displaystyle \mathbb{R}}$ at time ${\displaystyle t=1}$ is a function ${\displaystyle V:TM\to T^{2}M}$, which is an alternative description of the canonical vector field.

The existence of such a vector field on ${\displaystyle TM}$ is analogous to the canonical one-form on the cotangent bundle. Sometimes ${\displaystyle V}$ is also called the Liouville vector field, or radial vector field. Using ${\displaystyle V}$ one can characterize the tangent bundle. Essentially, ${\displaystyle V}$ 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 ${\displaystyle M}$ into objects on ${\displaystyle TM}$. For example, if ${\displaystyle \gamma }$ is a curve in ${\displaystyle M}$, then ${\displaystyle {\gamma }^{\prime }}$ (the tangent of ${\displaystyle \gamma }$) is a curve in ${\displaystyle TM}$. In contrast, without further assumptions on ${\displaystyle M}$ (say, a Riemannian metric), there is no similar lift into the cotangent bundle.

The vertical lift of a function ${\displaystyle f:M\to \mathbb{R}}$ is the function ${\displaystyle f^{\vee }:TM\to \mathbb{R}}$ defined by ${\displaystyle f^{\vee }=f\circ \pi }$, where ${\displaystyle \pi :TM\to M}$ is the canonical projection.