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 ${\displaystyle f:M\rightarrow N}$  is a smooth function, with ${\displaystyle M}$  and ${\displaystyle N}$  smooth manifolds, its derivative is a smooth function ${\displaystyle Df:TM\rightarrow TN}$ .

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 ${\displaystyle (U_{\alpha },\phi _{\alpha })}$ , where ${\displaystyle U_{\alpha }}$  is an open set in ${\displaystyle M}$  and

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

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

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

by

${\displaystyle {\widetilde {\phi }}_{\alpha }\left(x,v^{i}\partial _{i}\right)=\left(\phi _{\alpha }(x),v^{1},\cdots ,v^{n}\right)}$ 

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

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

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.

The simplest example is that of ${\displaystyle \mathbb {R} ^{n}}$ . In this case the tangent bundle is trivial: each ${\displaystyle T_{x}\mathbf {\mathbb {R} } ^{n}}$  is canonically isomorphic to ${\displaystyle T_{0}\mathbb {R} ^{n}}$  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.

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

${\displaystyle V\colon M\to TM}$ 

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 ${\displaystyle \Gamma (TM)}$ . 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 ${\displaystyle \Gamma (TM)}$  then takes on the structure of a module over the commutative algebra of smooth functions on M, denoted ${\displaystyle C^{\infty }(M)}$ .

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 ${\displaystyle \omega \in \Gamma (T^{*}M)}$ , ${\displaystyle \omega :M\to T^{*}M}$  that associate to each point ${\displaystyle x\in M}$  a 1-covector ${\displaystyle \omega _{x}\in T_{x}^{*}M}$ , which map tangent vectors to real numbers: ${\displaystyle \omega _{x}:T_{x}M\to \mathbb {R} }$ . Equivalently, a differential 1-form ${\displaystyle \omega \in \Gamma (T^{*}M)}$  maps a smooth vector field ${\displaystyle X\in \Gamma (TM)}$  to a smooth function ${\displaystyle \omega (X)\in C^{\infty }(M)}$ .

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 ${\displaystyle f:M\rightarrow N}$  has an induced derivative, for which the tangent bundle is the appropriate domain and range ${\displaystyle Df:TM\rightarrow TN}$ . 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.

On every tangent bundle ${\displaystyle TM}$ , considered as a manifold itself, one can define a canonical vector field ${\displaystyle V:TM\rightarrow T^{2}M}$  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, ${\displaystyle TW\cong W\times W,}$  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

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

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:

${\displaystyle {\begin{cases}\mathbb {R} \times TM\to TM\\(t,v)\longmapsto tv\end{cases}}}$ 

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

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 ${\displaystyle \gamma '}$  (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 ${\displaystyle f:M\rightarrow \mathbb {R} }$  is the function ${\displaystyle f^{\vee }:TM\rightarrow \mathbb {R} }$  defined by ${\displaystyle f^{\vee }=f\circ \pi }$ , where ${\displaystyle \pi :TM\rightarrow M}$  is the canonical projection.

• Pushforward (differential)
• Unit tangent bundle
• Cotangent bundle
• Frame bundle
• Musical isomorphism

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• Wolfram MathWorld: Tangent Bundle
• PlanetMath: Tangent Bundle