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{align} TM &= \bigsqcup_{x \in M} T_xM \\ &= \bigcup_{x \in M} \left\{x\right\} \times T_xM \\ &= \bigcup_{x \in M} \left\{(x, y) \mid y \in T_xM\right\} \\ &= \left\{ (x, y) \mid x \in M,\, y \in T_xM \right\} \end{align} $$

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 $$ T_xM$$ 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 Sn 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 $$ T_xU$$ 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 $$ T_xM\rightarrow\mathbb R^n$$ 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 $$V_x\in T_xM$$ 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 $$\omega_x \in T^*_xM$$, which map tangent vectors to real numbers: $$\omega_x : T_xM \to \R$$. 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 $$V:TM\rightarrow T^2M $$ 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 $$ V:TM\rightarrow T^2M $$, 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.