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tangent bundle
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{{Use American English|date = March 2019}}{{Short description|Tangent spaces of a manifold considered together}}right|thumb|Informally, the tangent bundle of a manifold (in this case a circle) is obtained by considering all the tangent spaces (top), and joining them together in a smooth and non-overlapping manner (bottom).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 unionThe 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. of the tangent spaces of M . That is,
begin{align}
TM &= bigsqcup_{x in M} T_xM
&= bigcup_{x in M} left{xright} times T_xM
&= bigcup_{x in M} left{(x, y) mid y in T_xMright}
&= left{ (x, y) mid x in M,, y in T_xM right}
end{align}
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 tangent space T_xM 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 (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 TMoplus E is trivial. For example, the n-dimensional sphere Sn is framed for all n, but parallelizable only for {{nowrap|1=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:Mrightarrow N is a smooth function, with M and N smooth manifolds, its derivative is a smooth function Df:TMrightarrow 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 Uis an open contractible subset of M, then there is a diffeomorphism TUrightarrow Utimesmathbb R^n which restricts to a linear isomorphism from each tangent space T_xU to {x}timesmathbb R^n. As a manifold, however, TM is not always diffeomorphic to the product manifold Mtimesmathbb R^n. When it is of the form Mtimesmathbb 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 Utimesmathbb 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_alphacolon U_alpha to mathbb R^n
is a diffeomorphism. These local coordinates on U give rise to an isomorphism T_xMrightarrowmathbb R^n forall xin U. We may then define a map
widetildephi_alphacolon pi^{-1}left(U_alpharight) to mathbb R^{2n}
by
widetildephi_alphaleft(x, v^ipartial_iright) = left(phi_alpha(x), v^1, cdots, v^nright)
We use these maps to define the topology and smooth structure on TM. A subset A of TM is open if and only if
widetildephi_alphaleft(Acap pi^{-1}left(U_alpharight)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_betaright) 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 Tmathbb 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^1timesmathbb 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
Vcolon M to TM
such that the image of x , denoted V_x , lies in T_xM , the tangent space at x . 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 Usubset 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^*Mthat associate to each point x in Ma 1-covector omega_x in T^*_xM, which map tangent vectors to real numbers: omega_x : T_xM to 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^text{th} order tangent bundle T^k M can be defined recursively as Tleft(T^{k-1}Mright).A smooth map f:Mrightarrow N has an induced derivative, for which the tangent bundle is the appropriate domain and range Df:TMrightarrow 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:TMrightarrow T^2M 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}^ntimesmathbb{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 tvend{cases}The derivative of this function with respect to the variable mathbb R at time t=1 is a function V:TMrightarrow T^2M , 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:Mrightarrowmathbb R is the function f^vee:TMrightarrowmathbb R defined by f^vee=fcirc pi, where pi:TMrightarrow M is the canonical projection.

See also

Notes

References

{{refimprove|date=July 2009}}{{Reflist}}
  • {{citation|first=Jeffrey M.|last=Lee|title=Manifolds and Differential Geometry|series=Graduate Studies in Mathematics|volume=Vol. 107 |publisher=American Mathematical Society|publication-place=Providence|year=2009}}. {{isbn|978-0-8218-4815-9}}
  • John M. Lee, Introduction to Smooth Manifolds, (2003) Springer-Verlag, New York. {{isbn|0-387-95495-3}}.
  • 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}}
  • M. De León, E. Merino, J.A. Oubiña, M. Salgado, A characterization of tangent and stable tangent bundles, Annales de l'institut Henri Poincaré (A) Physique théorique, Vol. 61, no. 1, 1994, 1-15 weblink

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