Double tangent bundle

Since (TM,π_TM,M) is a vector bundle in its own right, its tangent bundle has the secondary vector bundle structure (TTM,(π_TM)_*,TM), where (π_TM)_*:TTM→TM is the push-forward of the canonical projection π_TM:TM→M. In the following we denote

${\displaystyle \xi =\xi ^{k}{\frac {\partial }{\partial x^{k}}}{\Big |}_{x}\in T_{x}M,\qquad X=X^{k}{\frac {\partial }{\partial x^{k}}}{\Big |}_{x}\in T_{x}M}$

and apply the associated coordinate system

${\displaystyle \xi \mapsto (x^{1},\ldots ,x^{n},\xi ^{1},\ldots ,\xi ^{n})}$

on TM. Then the fibre of the secondary vector bundle structure at X∈T_xM takes the form

${\displaystyle (\pi _{TM})_{*}^{-1}(X)={\Big \{}\ X^{k}{\frac {\partial }{\partial x^{k}}}{\Big |}_{\xi }+Y^{k}{\frac {\partial }{\partial \xi ^{k}}}{\Big |}_{\xi }\ {\Big |}\ \xi \in T_{x}M\ ,\ Y^{1},\ldots ,Y^{n}\in \mathbb {R} \ {\Big \}}.}$

The double tangent bundle is a double vector bundle.

The canonical flip^[2] is a smooth involution j:TTM→TTM that exchanges these vector space structures in the sense that it is a vector bundle isomorphism between (TTM,π_TTM,TM) and (TTM,(π_TM)_*,TM). In the associated coordinates on TM it reads as

${\displaystyle j{\Big (}X^{k}{\frac {\partial }{\partial x^{k}}}{\Big |}_{\xi }+Y^{k}{\frac {\partial }{\partial \xi ^{k}}}{\Big |}_{\xi }{\Big )}=\xi ^{k}{\frac {\partial }{\partial x^{k}}}{\Big |}_{X}+Y^{k}{\frac {\partial }{\partial \xi ^{k}}}{\Big |}_{X}.}$

The canonical flip has the property that for any f: R² → M,

${\displaystyle {\frac {\partial f}{{\partial t}{\partial s}}}=j\circ {\frac {\partial f}{{\partial s}{\partial t}}}}$

where s and t are coordinates of the standard basis of R ². Note that both partial derivatives are functions from R² to TTM.

This property can, in fact, be used to give an intrinsic definition of the canonical flip.^[3] Indeed, there is a submersion p: J²₀ (R²,M) → TTM given by

${\displaystyle p([f])={\frac {\partial f}{{\partial t}{\partial s}}}(0,0)}$

where p can be defined in the space of two-jets at zero because only depends on f up to order two at zero. We consider the application:

${\displaystyle J:J_{0}^{2}(\mathbb {R} ^{2},M)\to J_{0}^{2}(\mathbb {R} ^{2},M)\quad /\quad J([f])=[f\circ \alpha ]}$

where α(s,t)= (t,s). Then J is compatible with the projection p and induces the canonical flip on the quotient TTM.

Summarize Timeline Top Qs Fact Check

As for any vector bundle, the tangent spaces T_ξ(T_xM) of the fibres T_xM of the tangent bundle (TM,π_TM,M) can be identified with the fibres T_xM themselves. Formally this is achieved through the vertical lift, which is a natural vector space isomorphism vl_ξ:T_xM→V_ξ(T_xM) defined as

${\displaystyle (\operatorname {vl} _{\xi }X)[f]:={\frac {d}{dt}}{\Big |}_{t=0}f(x,\xi +tX),\qquad f\in C^{\infty }(TM).}$

The vertical lift can also be seen as a natural vector bundle isomorphism vl:(π_TM)^*TM→VTM from the pullback bundle of (TM,π_TM,M) over π_TM:TM→M onto the vertical tangent bundle

${\displaystyle VTM:=\operatorname {Ker} (\pi _{TM})_{*}\subset TTM.}$

The vertical lift lets us define the canonical vector field

${\displaystyle V:TM\to TTM;\qquad V_{\xi }:=\operatorname {vl} _{\xi }\xi ,}$

which is smooth in the slit tangent bundle TM\0. The canonical vector field can be also defined as the infinitesimal generator of the Lie-group action

${\displaystyle \mathbb {R} \times (TM\setminus 0)\to TM\setminus 0;\qquad (t,\xi )\mapsto e^{t}\xi .}$

Unlike the canonical vector field, which can be defined for any vector bundle, the canonical endomorphism

${\displaystyle J:TTM\to TTM;\qquad J_{\xi }X:=\operatorname {vl} _{\xi }(\pi _{TM})_{*}X,\qquad X\in T_{\xi }TM}$

is special to the tangent bundle. The canonical endomorphism J satisfies

${\displaystyle \operatorname {Ran} (J)=\operatorname {Ker} (J)=VTM,\qquad {\mathcal {L}}_{V}J=-J,\qquad [JX,JY]=J[JX,Y]+J[X,JY],}$

and it is also known as the tangent structure for the following reason. If (E,p,M) is any vector bundle with the canonical vector field V and a (1,1)-tensor field J that satisfies the properties listed above, with VE in place of VTM, then the vector bundle (E,p,M) is isomorphic to the tangent bundle (TM,π_TM,M) of the base manifold, and J corresponds to the tangent structure of TM in this isomorphism.

There is also a stronger result of this kind ^[4] which states that if N is a 2n-dimensional manifold and if there exists a (1,1)-tensor field J on N that satisfies

${\displaystyle \operatorname {Ran} (J)=\operatorname {Ker} (J),\qquad [JX,JY]=J[JX,Y]+J[X,JY],}$

then N is diffeomorphic to an open set of the total space of a tangent bundle of some n-dimensional manifold M, and J corresponds to the tangent structure of TM in this diffeomorphism.

In any associated coordinate system on TM the canonical vector field and the canonical endomorphism have the coordinate representations

${\displaystyle V=\xi ^{k}{\frac {\partial }{\partial \xi ^{k}}},\qquad J=dx^{k}\otimes {\frac {\partial }{\partial \xi ^{k}}}.}$

A Semispray structure on a smooth manifold M is by definition a smooth vector field H on TM \0 such that JH=V. An equivalent definition is that j(H)=H, where j:TTM→TTM is the canonical flip. A semispray H is a spray, if in addition, [V,H]=H.

Spray and semispray structures are invariant versions of second order ordinary differential equations on M. The difference between spray and semispray structures is that the solution curves of sprays are invariant in positive reparametrizations^[jargon] as point sets on M, whereas solution curves of semisprays typically are not.