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 $(TE,E,TM,M)$.

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 $2N$, 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 $φ(x) = (x^{1}, ..., x^{n})$ 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 $T_{x}M$ 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 $χ : TW → TU × R^{2N}$ 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 $TE = HE ⊕ VE$ 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 $vl_{v} : E → V_{v}E$ is the vertical lift, and $vpr_{v} : T_{v}E → V_{v}E$ is the vertical projection. The mapping


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

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 $(TE, π_{TE}, E)$.