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^{[note 1]} 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 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 $T_{x}U$ 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.

Notes

^ ^a ^b The disjoint union ensures that for any two points $x 1$ and $x 2$ of manifold $M$ the tangent spaces $T 1$ and $T 2$ have no common vector. This is graphically illustrated in the accompanying picture for tangent bundle of circle $S 1$ , see Examples section: all tangents to a circle lie in the plane of the circle. In order to make them disjoint it is necessary to align them in a plane perpendicular to the plane of the circle.

References

Lee, Jeffrey M. (2009), Manifolds and Differential Geometry, Graduate Studies in Mathematics, vol. 107, Providence: American Mathematical Society. ISBN 978-0-8218-4815-9
Lee, John M. (2012). Introduction to Smooth Manifolds. Graduate Texts in Mathematics. Vol. 218. doi:10.1007/978-1-4419-9982-5. ISBN 978-1-4419-9981-8.
Jürgen Jost, Riemannian Geometry and Geometric Analysis, (2002) Springer-Verlag, Berlin. ISBN 3-540-42627-2
Ralph Abraham and Jerrold E. Marsden, Foundations of Mechanics, (1978) Benjamin-Cummings, London. ISBN 0-8053-0102-X
León, M. De; Merino, E.; Oubiña, J. A.; Salgado, M. (1994). "A characterization of tangent and stable tangent bundles" (PDF). Annales de l'I.H.P.: Physique Théorique. 61 (1): 1–15.
Gudmundsson, Sigmundur; Kappos, Elias (2002). "On the geometry of tangent bundles". Expositiones Mathematicae. 20: 1–41. doi:10.1016/S0723-0869(02)80027-5.

External links

"Tangent bundle", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
Wolfram MathWorld: Tangent Bundle
PlanetMath: Tangent Bundle

[disjoint-1] The disjoint union ensures that for any two points $x 1$ and $x 2$ of manifold $M$ the tangent spaces $T 1$ and $T 2$ have no common vector. This is graphically illustrated in the accompanying picture for tangent bundle of circle $S 1$ , see Examples section: all tangents to a circle lie in the plane of the circle. In order to make them disjoint it is necessary to align them in a plane perpendicular to the plane of the circle.

[note 1]