# 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 : TETM of the original projection map p : EM. This gives rise to a double vector bundle structure .

In the special case (E, p, M) = (TM, πTM, M), where TE = TTM is the double tangent bundle, the secondary vector bundle (TTM, (πTM), TM) is isomorphic to the tangent bundle (TTM, πTTM, 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 : TETM of the canonical projection p : EM is a smooth submanifold of dimension 2N, 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 (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) = (x1, ..., xn) and let

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

be a coordinate system on $W:=p^{-1}(U)\subset E$ adapted to it. Then

$p_*\left (X^k\frac{\partial}{\partial x^k}\Bigg|_v + Y^\ell\frac{\partial}{\partial v^\ell}\Bigg|_v \right) = X^k\frac{\partial}{\partial x^k}\Bigg|_{p(v)},$

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

$p^{-1}_*(X) = \left \{ X^k\frac{\partial}{\partial x^k}\Bigg|_v + Y^\ell\frac{\partial}{\partial v^\ell}\Bigg|_v \ : \ v\in E_x; Y^1,\ldots,Y^N\in\mathbf{R} \right \}.$

Now it turns out that

$\chi\left(X^k\frac{\partial}{\partial x^k}\Bigg|_v + Y^\ell\frac{\partial}{\partial v^\ell}\Bigg|_v\right ) = \left (X^k\frac{\partial}{\partial x^k}\Bigg|_{p(v)}, \left (v^1,\ldots,v^N,Y^1,\ldots,Y^N \right) \right )$

gives a local trivialization χ : TWTU × R2N for (TE, p, TM), and the push-forwards of the original vector space operations read in the adapted coordinates as

$\left (X^k\frac{\partial}{\partial x^k}\Bigg|_v + Y^\ell\frac{\partial}{\partial v^\ell}\Bigg|_v\right) +_* \left (X^k\frac{\partial}{\partial x^k}\Bigg|_w + Z^\ell\frac{\partial}{\partial v^\ell}\Bigg|_w\right) = X^k\frac{\partial}{\partial x^k}\Bigg|_{v+w} + (Y^\ell+Z^\ell)\frac{\partial}{\partial v^\ell}\Bigg|_{v+w}$

and

$\lambda_*\left (X^k\frac{\partial}{\partial x^k}\Bigg|_v + Y^\ell\frac{\partial}{\partial v^\ell}\Bigg|_v\right) = X^k\frac{\partial}{\partial x^k}\Bigg|_{\lambda v} + \lambda Y^\ell\frac{\partial}{\partial v^\ell}\Bigg|_{\lambda v},$

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 = HEVE on a vector bundle (E, p, M) can be characterized in terms of the connector map

$\begin{cases}\kappa:T_vE\to E_{p(v)} \\ \kappa(X):=\operatorname{vl}_v^{-1}(\operatorname{vpr}X) \end{cases}$

where vlv : EVvE is the vertical lift, and vprv : TvEVvE is the vertical projection. The mapping

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

induced by an Ehresmann connection is a covariant derivative on Γ(E) in the sense that

\begin{align} \nabla_{X+Y}v &= \nabla_X v + \nabla_Y v \\ \nabla_{\lambda X}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]v + f\nabla_Xv \end{align}

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, πTE, E).