# Double tangent bundle

In mathematics, particularly differential topology, the **double tangent bundle** or the **second tangent bundle** refers to the tangent bundle *T*^{2}*M*=*T*(*TM*) of the tangent bundle *TM* of a smooth manifold *M*. The double tangent bundle arises in the study of connections and second order ordinary differential equations, i.e. (semi)spray structures on smooth manifolds.

The double tangent bundle is closely related to the second order jet bundle, which is an object specifically designed to hold the "2nd order derivative information" of smooth functions on smooth manifolds.

Given a smooth map there is an induced 1st-order derivative map and so also a 2nd order derivative map .

The Lie Bracket of two vector fields on a manifold also has a formulation in terms of the double tangent bundle.

## Contents

## Canonical flip and associated coordinates

Unlike the tangent bundle, which only has one canonical vector space structure,
the double tangent bundle has *two* vector bundle structures. Suppose and are canonical projections for and , respectively. Then has vector bundle structures and .
The canonical *flip map* is a diffeomorphism that exchanges these vector space structures.

Let us give a coordinate representation for the canonical flip. Starting from local coordinates (*U*,φ) of the base manifold *M* we define the associated coordinates (*TU*,*T*φ) on *TM* by

In terms of the coodinate basis of these associated coordinates the canonical flip reads as

Let be the local coordinates on associated with the local coordinates on *TM* associated with the local coordinates on *M*. Then the canonical flip reads as

## Canonical tensor fields on the tangent bundle

As for any vector bundle, the tangent spaces *T*_{v}(*T*_{x}*M*) of the fibres *T*_{x}*M* of (*TM*,π,*M*) can be identified with the fibres *T*_{x}*M* themselves. Formally this is achieved though the **vertical lift**

The vertical lift is a natural vector bundle isomorphism *vl*:π^{*}*TM*→*VTM* from the pullback bundle of (*TM*,π,*M*) over π:*TM*→*M* onto the vertical tangent bundle *VTM*:=*Ker*(π_{*})⊂*TTM*.

In terms of the vertical lift we can define two canonical tensor fields on (*TTM*,π_{2},*TM*), the **canonical vector field**

and the **tangent structure** or **canonical endomorphism**

In the local coordinates (*TU*,*T*φ) on *TM* associated to (*U*,φ)these canonical tensor fields have coordinate representations

The canonical endomorphism *J* satisfies

and in a certain sense the existence of such tensor field *J* on a 2*n*-dimensional manifold implies that the manifold is a (part of) a tangent bundle of some *n*-dimensional manifold.

The canonical vector field can also be defined as follows. For *v*∈*TM* define *f*_{v}:**R**→*TM* by *f*_{v}(*t*)=*tv*, where *tv*∈*TM* is the scalar multiplication. Then *V*_{v}:=*Tf*_{v}(1,1)∈*TTM*, where we identify *T***R** with **R**^{2} in the standard way.

## (Semi)spray structures

A Semispray structure on a smooth manifold *M* is dy definition a smooth vector field *H* on *TM* \0, or in other words, a smooth section of the deleted double tangent bundle (*T*(*TM* \0),π_{2},*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 ivariant 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 as point sets on *M*, whereas solution curves of semisprays typically are not.

## Connections on the tangent bundle

Connections on has a formulation in terms of a *projection map* due to Ehresmann.
Let and be the bundle projections for the double tangent bundle and tangent bundle respectively.

## See also

- Spray (mathematics)

## References

- D.S.Goel.
*Almost tangent structures,*Kodai.Math.Sem.Rep.**26**(1975) 187-193. - P.Michor.
*Topics in Differential Geometry,*American Mathematical Society (2008). - Guillemin and Pollack.
*Differential Topology,*Prentice-Hall (1974).

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