Tangent bundle

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).

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 ${\displaystyle M}$ is a manifold ${\displaystyle TM}$ which assembles all the tangent vectors in ${\displaystyle M}$. As a set, it is given by the disjoint union ^{[note 1]} of the tangent spaces of ${\displaystyle M}$. That is,

${\displaystyle {\begin{aligned}TM&=\bigsqcup _{x\in M}T_{x}M\\&=\bigcup _{x\in M}\left\{x\right\}\times T_{x}M\\&=\bigcup _{x\in M}\left\{(x,y)\mid y\in T_{x}M\right\}\\&=\left\{(x,y)\mid x\in M,\,y\in T_{x}M\right\}\end{aligned}}}$

where ${\displaystyle T_{x}M}$ denotes the tangent space to ${\displaystyle M}$ at the point ${\displaystyle x}$. So, an element of ${\displaystyle TM}$ can be thought of as a pair ${\displaystyle (x,v)}$, where ${\displaystyle x}$ is a point in ${\displaystyle M}$ and ${\displaystyle v}$ is a tangent vector to ${\displaystyle M}$ at ${\displaystyle x}$.

There is a natural projection

${\displaystyle \pi :TM\twoheadrightarrow M}$

defined by ${\displaystyle \pi (x,v)=x}$. This projection maps each element of the tangent space ${\displaystyle T_{x}M}$ to the single point ${\displaystyle x}$.

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 ${\displaystyle TM}$ is a vector field on ${\displaystyle M}$, and the dual bundle to ${\displaystyle TM}$ is the cotangent bundle, which is the disjoint union of the cotangent spaces of ${\displaystyle M}$. By definition, a manifold ${\displaystyle M}$ is parallelizable if and only if the tangent bundle is trivial. By definition, a manifold ${\displaystyle M}$ is framed if and only if the tangent bundle ${\displaystyle TM}$ is stably trivial, meaning that for some trivial bundle ${\displaystyle E}$ the Whitney sum ${\displaystyle TM\oplus E}$ 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 ${\displaystyle f:M\rightarrow N}$ is a smooth function, with ${\displaystyle M}$ and ${\displaystyle N}$ smooth manifolds, its derivative is a smooth function ${\displaystyle Df:TM\rightarrow TN}$.

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 ${\displaystyle TM}$ 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.

Examples

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.

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 ${\displaystyle M}$ 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 ${\displaystyle \omega (X)\in C^{\infty }(M)}$.

Higher-order tangent bundles

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.

Canonical vector field on tangent bundle

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.

Lifts

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.

See also

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

Notes

1. The disjoint union ensures that for any two points x₁ and x₂ of manifold M the tangent spaces T₁ and T₂ have no common vector. This is graphically illustrated in the accompanying picture for tangent bundle of circle S¹, 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