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Tangent bundle - Wikipedia

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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).[note 1]

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).

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

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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)}.

Higher-order tangent bundles

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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

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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.

  1. ^ a b 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.