In mathematics, particularly differential topology, the secondary vector bundle structurerefers 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.
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 x E)\toTE, λ*:TE\toTE
of the original addition and scalar multiplication
+:E x E\toE, λ:E\toE
as its vector space operations. The triple becomes a smooth vector bundle with these vector space operations on its fibres.
Let be a local coordinate system on the base manifold with and let
\begin{cases}\psi:W\to\varphi(U) x RN\ \psi\left(vkek|x\right):=\left(x1,\ldots,xn,v1,\ldots,vN\right)\end{cases}
be a coordinate system on
W:=p-1(U)\subsetE
p*\left
| ||||
(X |
|v+
| ||||
Y |
|v\right)=
| ||||
X |
|p(v),
so the fiber of the secondary vector bundle structure at in is of the form
-1 | |
p | |
*(X) |
=\left\{
| ||||
X |
|v+
| ||||
Y |
|v : v\inEx;Y1,\ldots,YN\inR\right\}.
Now it turns out that
| ||||
\chi\left(X |
|v+
| ||||
Y |
|v\right)=\left
| ||||
(X |
|p(v),\left(v1,\ldots,vN,Y1,\ldots,YN\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 |
|v+
| ||||
Y |
|v\right)+*\left
| ||||
(X |
|w+
| ||||
Z |
|w\right)=
| ||||
X |
|v+w+(Y\ell+Z
| ||||
|v+w
and
λ*\left
| ||||
(X |
|v+
| ||||
Y |
|v\right)=
| ||||
X |
|λ+λ
| ||||
Y |
|λ,
so each fibre is a vector space and the triple is a smooth vector bundle.
The general Ehresmann connection on a vector bundle can be characterized in terms of the connector map
\begin{cases}\kappa:TvE\toEp(v)
-1 | |
\ \kappa(X):=\operatorname{vl} | |
v |
(\operatorname{vpr}X)\end{cases}
where is the vertical lift, and is the vertical projection. The mapping
\begin{cases}\nabla:\Gamma(TM) x \Gamma(E)\to\Gamma(E)\ \nablaXv:=\kappa(v*X)\end{cases}
induced by an Ehresmann connection is a covariant derivative on in the sense that
\begin{align} \nablaX+Yv&=\nablaXv+\nablaYv\\ \nablaλv&=λ\nablaXv\\ \nablaX(v+w)&=\nablaXv+\nablaXw\\ \nablaX(λv)&=λ\nablaXv\\ \nablaX(fv)&=X[f]v+f\nablaXv \end{align}
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 .