# Secondary vector bundle structure

In mathematics, particularly differential topology, the **secondary vector bundle structure**
refers to the natural vector bundle structure (*TE*, *p*_{∗}, *TM*) on the total space *TE* of the tangent bundle of a smooth vector bundle (*E*, *p*, *M*), induced by the push-forward *p*_{∗} : *TE* → *TM* of the original projection map *p* : *E* → *M*.
This gives rise to a double vector bundle structure .

In the special case (*E*, *p*, *M*) = (*TM*, *π _{TM}*,

*M*), where

*TE*=

*TTM*is the double tangent bundle, the secondary vector bundle (

*TTM*, (

*π*)

_{TM}_{∗},

*TM*) is isomorphic to the tangent bundle (

*TTM*,

*π*,

_{TTM}*TM*) of

*TM*through the canonical flip.

## Construction of the secondary vector bundle structure

Let (*E*, *p*, *M*) be a smooth vector bundle of rank N. Then the preimage (*p*_{∗})^{−1}(*X*) ⊂ *TE* of any tangent vector X in *TM* in the push-forward *p*_{∗} : *TE* → *TM* of the canonical projection *p* : *E* → *M* is a smooth submanifold of dimension 2*N*, and it becomes a vector space with the push-forwards

of the original addition and scalar multiplication

as its vector space operations. The triple (*TE*, *p*_{∗}, *TM*) becomes a smooth vector bundle with these vector space operations on its fibres.

### Proof

Let (*U*, *φ*) be a local coordinate system on the base manifold M with *φ*(*x*) = (*x*^{1}, ..., *x ^{n}*) and let

be a coordinate system on adapted to it. Then

so the fiber of the secondary vector bundle structure at X in *T _{x}M* is of the form

Now it turns out that

gives a local trivialization *χ* : *TW* → *TU* × **R**^{2N} for (*TE*, *p*_{∗}, *TM*), and the push-forwards of the original vector space operations read in the adapted coordinates as

and

so each fibre (*p*_{∗})^{−1}(*X*) ⊂ *TE* is a vector space and the triple (*TE*, *p*_{∗}, *TM*) is a smooth vector bundle.

## Linearity of connections on vector bundles

The general Ehresmann connection *TE* = *HE* ⊕ *VE* on a vector bundle (*E*, *p*, *M*) can be characterized in terms of the **connector map**

where vl_{v} : *E* → *V _{v}E* is the vertical lift, and vpr

_{v}:

*T*→

_{v}E*V*is the vertical projection. The mapping

_{v}Einduced by an Ehresmann connection is a covariant derivative on Γ(*E*) in the sense that

if and only if the connector map is linear with respect to the secondary vector bundle structure (*TE*, *p*_{∗}, *TM*) on *TE*. Then the connection is called *linear*. Note that the connector map is automatically linear with respect to the tangent bundle structure (*TE*, *π _{TE}*,

*E*).

## See also

## References

- P.Michor.
*Topics in Differential Geometry,*American Mathematical Society (2008).