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In differential geometry, the tangent bundle of a differentiable manifold M is the disjoint union[note 1] of the tangent spaces of M. That is,

\( TM = \bigsqcup_{x\in M}T_xM=\bigcup_{x\in M} \left\{x\right\}\times T_xM. \)

where TxM 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 π(x, v) = x. This projection maps each tangent space TxM to the single point x.

The tangent bundle to a manifold is the prototypical example of a vector bundle (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 ⊕ 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).

Informally, the tangent bundle of a manifold (in this case a circle) is obtained by considering all the tangent spaces (top), and joining them together in a smooth and non-overlapping manner (bottom).[note 1]

Role

The main role of the tangent bundle is to provide a domain and range for the derivative of a smooth function. Namely, if f : M \to N is a smooth function, with M and N smooth manifolds, its derivative is a smooth function \( Df : TM \to TN \).

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 from TU to U × Rⁿ which restricts to a linear isomorphism from each tangent space T_xU to {x} × Rⁿ . As a manifold, however, TM is not always diffeomorphic to the product manifold M × Rⁿ. When it is of the form M × Rⁿ, 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 modelled on Euclidean space, tangent bundles are locally modelled on U × Rⁿ, 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_α, φ_α) where U_α is an open set in M and

\( \phi_\alpha\colon U_\alpha \to \mathbb R^n \)

is a diffeomorphism. These local coordinates on U give rise to an isomorphism between T_xM and Rⁿ for each x ∈ U. We may then define a map

\( \tilde\phi_\alpha\colon \pi^{-1}(U_\alpha) \to \mathbb R^{2n} \)

\( \tilde\phi_\alpha(x, v^i\partial_i) = (\phi_\alpha(x), v^1, \cdots, v^n) \)

We use these maps to define the topology and smooth structure on TM. A subset A of TM is open if and only if \( \tilde\phi_\alpha(A\cap \pi^{-1}(U_\alpha)) \) is open in R²ⁿ for each α. These maps are then homeomorphisms between open subsets of TM and R²ⁿ and therefore serve as charts for the smooth structure on TM. The transition functions on chart overlaps \( \pi^{-1}(U_\alpha\cap U_\beta) \) are induced by the Jacobian matrices of the associated coordinate transformation and are therefore smooth maps between open subsets of R²ⁿ.

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.

ExamplesA simple example of a nontrivial tangent bundle is that of the unit sphere S2: this tangent bundle is nontrivial as a consequence of the hairy ball theorem. Therefore, the sphere is not parallelizable.

The simplest example is that of Rⁿ. In this case the tangent bundle is trivial.

Another simple example is the unit circle, S¹ (see picture above). The tangent bundle of the circle is also trivial and isomorphic to S¹ × R. Geometrically, this is a cylinder of infinite height.

The only tangent bundles that can be readily visualized are those of the real line R and the unit circle S¹, 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 S²: 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 the image of x, denoted Vx, lies in TxM, the tangent space at x. 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 Γ(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 Γ(TM) then takes on the structure of a module over the commutative algebra of smooth functions on M, denoted C∞(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 in 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.
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 kth-order tangent bundle T^k M can be defined recursively as T(T^{k-1}M).

A smooth map f : M \to N has an induced derivative, for which the tangent bundle is the appropriate domain and range \( Df : TM \to TN \). 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 one can define a canonical vector field V : \( TM \to TTM \). If (x, v) are local coordinates for TM, the vector field has the expression

\( V = \sum_i \left. v^i \frac{\partial}{\partial v^i} \right|_{(x,v)}. \)

Alternatively, consider \mathbb R \times TM \to TM to be the scalar multiplication function (t,v) \longmapsto tv. The derivative of this function with respect to the variable \( \mathbb R \) at time t=1 is a function \( V : TM \to TTM \), which is an alternative description of the canonical vector field.

The existence of such a vector field on TM can be compared with the existence of a canonical 1-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 c is a curve in M, then c' (the tangent of c) is a curve in TM. Let us point out that without further assumptions on M (say, a Riemannian metric), there is no similar lift into the cotangent bundle.

The vertical lift of a function \( f\colon M\to \mathbb{R} \) is the function \( f^v\colon TM\to \mathbb{R} \) defined by \( f^v=f\circ \pi \) , where \( \pi\colon TM\to M \) is the canonical projection.

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Tangent bundle