secondary vector bundle structure

In mathematics, particularly differential topology, the secondary vector bundle structure
refers to the natural vector bundle structure on the total space ''TE'' of the tangent bundle of a smooth vector bundle , induced by the push-forward of the original projection map .
This gives rise to a double vector bundle structure .

In the special case , where is the double tangent bundle, the secondary vector bundle is isomorphic to the tangent bundle
of through the canonical flip.

Construction of the secondary vector bundle structure

Let be a smooth vector bundle of rank . Then the preimage of any tangent vector in in the push-forward of the canonical projection is a smooth submanifold of dimension , and it becomes a vector space with the push-forwards

:$+_*:T(E\times E)\to TE, \qquad \lambda_*:TE\to TE$

of the original addition and scalar multiplication

:$+:E\times E\to E, \qquad \lambda:E\to E$

as its vector space operations. The triple becomes a smooth vector bundle with these vector space operations on its fibres.

Proof

Let be a local coordinate system on the base manifold with and let

:$\begin\psi:W \to \varphi(U)\times \mathbf^N \\ \psi \left (v^k e_k, _x \right ) := \left (x^1,\ldots,x^n,v^1,\ldots,v^N \right )\end$

be a coordinate system on $W:=p^(U)\subset E$ adapted to it. Then

:$p_*\left (X^k\frac\Bigg, _v + Y^\ell\frac\Bigg, _v \right) = X^k\frac\Bigg, _,$

so the fiber of the secondary vector bundle structure at in is of the form

:$p^_*(X) = \left \.$

Now it turns out that

:$\chi\left(X^k\frac\Bigg, _v + Y^\ell\frac\Bigg, _v\right ) = \left (X^k\frac\Bigg, _, \left (v^1,\ldots,v^N,Y^1,\ldots,Y^N \right) \right )$

gives a local trivialization for , and the push-forwards of the original vector space operations read in the adapted coordinates as

:$\left (X^k\frac\Bigg, _v + Y^\ell\frac\Bigg, _v\right) +_* \left (X^k\frac\Bigg, _w + Z^\ell\frac\Bigg, _w\right) = X^k\frac\Bigg, _ + (Y^\ell+Z^\ell)\frac\Bigg, _$

and

:$\lambda_*\left (X^k\frac\Bigg, _v + Y^\ell\frac\Bigg, _v\right) = X^k\frac\Bigg, _ + \lambda Y^\ell\frac\Bigg, _,$

so each fibre is a vector space and the triple is a smooth vector bundle.

Linearity of connections on vector bundles

The general Ehresmann connection on a vector bundle can be characterized in terms of the connector map

:$\begin\kappa:T_vE\to E_ \\ \kappa(X):=\operatorname_v^(\operatornameX) \end$

where is the vertical lift, and is the vertical projection. The mapping

:$\begin\nabla:\Gamma(TM)\times\Gamma(E)\to\Gamma(E) \\ \nabla_Xv := \kappa(v_*X) \end$

induced by an Ehresmann connection is a covariant derivative on in the sense that

:\begin
\nabla_v &= \nabla_X v + \nabla_Y v \\
\nabla_v &=\lambda \nabla_Xv \\
\nabla_X(v+w) &= \nabla_X v + \nabla_X w \\
\nabla_X(\lambda v) &=\lambda \nabla_Xv \\
\nabla_X(fv) &= X + f\nabla_Xv
\end

if and only if the connector map is linear with respect to the secondary vector bundle structure on . Then the connection is called ''linear''. Note that the connector map is automatically linear with respect to the tangent bundle structure {{math, (''TE'', ''π_TE'', ''E'').

See also

* Connection (vector bundle)
* Double tangent bundle
* Ehresmann connection
* Vector bundle

References

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

Differential geometry
Topology
Differential topology