Tangent bundle

File:Tangent bundle.svg

Informally, the tangent bundle of a manifold (which in this case is a circle) is obtained by considering all the tangent spaces (top), and joining them together in a smooth and non-overlapping manner (bottom).<ref group=note name="disjoint"/>

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 <math> M </math> is a manifold <math>TM</math> which assembles all the tangent vectors in <math> M </math>. As a set, it is given by the disjoint union<ref group="note" name="disjoint">The disjoint union ensures that for any two points Template:Math and Template:Math of manifold Template:Math the tangent spaces Template:Math and Template:Math have no common vector. This is graphically illustrated in the accompanying picture for tangent bundle of circle Template:Math, 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.</ref> of the tangent spaces of <math> M </math>. That is,

<math>

\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}

</math>

where <math> T_x M</math> denotes the tangent space to <math> M </math> at the point <math> x </math>. So, an element of <math> TM</math> can be thought of as a pair <math> (x,v)</math>, where <math> x </math> is a point in <math> M </math> and <math> v </math> is a tangent vector to <math> M </math> at <math> x </math>.

There is a natural projection

<math> \pi : TM \twoheadrightarrow M </math>

defined by <math> \pi(x, v) = x</math>. This projection maps each element of the tangent space <math> T_xM</math> to the single point <math> x </math>.

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 <math> TM</math> is a vector field on <math> M</math>, and the dual bundle to <math> TM</math> is the cotangent bundle, which is the disjoint union of the cotangent spaces of <math> M </math>. By definition, a manifold <math> M </math> is parallelizable if and only if the tangent bundle is trivial. By definition, a manifold <math>M</math> is framed if and only if the tangent bundle <math>TM</math> is stably trivial, meaning that for some trivial bundle <math>E</math> the Whitney sum <math> TM\oplus E</math> is trivial. For example, the n-dimensional sphere Sⁿ is framed for all n, but parallelizable only for Template:Nowrap (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 <math> f:M\rightarrow N </math> is a smooth function, with <math> M </math> and <math> N </math> smooth manifolds, its derivative is a smooth function <math> Df:TM\rightarrow TN </math>.

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 <math> TM</math> is twice the dimension of <math> M</math>.

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

If M is a smooth n-dimensional manifold, then it comes equipped with an atlas of charts <math>(U_\alpha,\phi_\alpha)</math>, where <math> U_\alpha</math> is an open set in <math>M</math> and

<math>\phi_\alpha: U_\alpha \to \mathbb R^n</math>

is a diffeomorphism. These local coordinates on <math> U_\alpha </math> give rise to an isomorphism <math> T_xM\rightarrow\mathbb R^n</math> for all <math> x\in U_\alpha</math>. We may then define a map

<math>\widetilde\phi_\alpha:\pi^{-1}\left(U_\alpha\right) \to \mathbb R^{2n}</math>

by

<math>\widetilde\phi_\alpha\left(x, v^i\partial_i\right) = \left(\phi_\alpha(x), v^1, \cdots, v^n\right)</math>

We use these maps to define the topology and smooth structure on <math>TM</math>. A subset <math>A</math> of <math> TM</math> is open if and only if

<math>\widetilde\phi_\alpha\left(A\cap \pi^{-1}\left(U_\alpha\right)\right)</math>

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

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 <math>n</math>-dimensional manifold <math>M</math> may be defined as a rank <math>n</math> vector bundle over <math>M</math> whose transition functions are given by the Jacobian of the associated coordinate transformations.

Examples

The simplest example is that of <math>\mathbb R^n</math>. In this case the tangent bundle is trivial: each <math> T_x \mathbf \mathbb R^n </math> is canonically isomorphic to <math> T_0 \mathbb R^n </math> via the map <math> \mathbb R^n \to \mathbb R^n </math> which subtracts <math> x </math>, giving a diffeomorphism <math> T\mathbb R^n \to \mathbb R^n \times \mathbb R^n</math>.

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

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

<math>V\colon M \to TM</math>

such that <math>V(x) = (x,V_x)</math> with <math>V_x\in T_xM</math> for every <math>x\in M</math>. In the language of fiber bundles, such a map is called a section. A vector field on <math>M</math> is therefore a section of the tangent bundle of <math>M</math>.

The set of all vector fields on <math>M</math> is denoted by <math>\Gamma(TM)</math>. Vector fields can be added together pointwise

<math>(V+W)_x = V_x + W_x</math>

and multiplied by smooth functions on M

<math>(fV)_x = f(x)V_x</math>

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

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

The above construction applies equally well to the cotangent bundle – the differential 1-forms on <math>M</math> are precisely the sections of the cotangent bundle <math>\omega \in \Gamma(T^*M)</math>, <math>\omega: M \to T^*M</math> that associate to each point <math>x \in M</math> a 1-covector <math>\omega_x \in T^*_xM</math>, which map tangent vectors to real numbers: <math>\omega_x : T_xM \to \R</math>. Equivalently, a differential 1-form <math>\omega \in \Gamma(T^*M)</math> maps a smooth vector field <math>X \in \Gamma(TM)</math> to a smooth function <math>\omega(X) \in C^{\infty}(M)</math>.

Higher-order tangent bundles

Since the tangent bundle <math>TM</math> is itself a smooth manifold, the second-order tangent bundle can be defined via repeated application of the tangent bundle construction:

<math>T^2 M = T(TM).\,</math>

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

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

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

<math>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)</math>

and the map <math>TTM \to TM</math> is the projection onto the first coordinates:

<math>(TM \to M) \times (\mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}^n).</math>

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

If <math>(x,v)</math> are local coordinates for <math>TM</math>, the vector field has the expression

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

More concisely, <math>(x, v) \mapsto (x, v, 0, v)</math> – 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 <math>v</math>, not on <math>x</math>, as only the tangent directions can be naturally identified.

Alternatively, consider the scalar multiplication function:

<math>\begin{cases}

\mathbb{R} \times TM \to TM \\ (t,v) \longmapsto tv \end{cases}</math>

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

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

The vertical lift of a function <math> f:M\rightarrow\mathbb R </math> is the function <math> f^\vee:TM\rightarrow\mathbb R </math> defined by <math>f^\vee=f\circ \pi</math>, where <math> \pi:TM\rightarrow M </math> is the canonical projection.

See also

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

Notes

<references group=note/>

References

• Template:Citation. Template:Isbn
• Template:Cite book
• Jürgen Jost, Riemannian Geometry and Geometric Analysis, (2002) Springer-Verlag, Berlin. Template:Isbn
• Ralph Abraham and Jerrold E. Marsden, Foundations of Mechanics, (1978) Benjamin-Cummings, London. Template:Isbn
• Template:Cite journal
• Template:Cite journal

External links

• Template:Springer
• Wolfram MathWorld: Tangent Bundle
• PlanetMath: Tangent Bundle