Secondary vector bundle structure

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

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

Construction of the secondary vector bundle structure

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

${+}_{*}:T(E\times E)\to TE,{\lambda }_{*}:TE\to TE$

of the original addition and scalar multiplication

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

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

Proof

Let Template:Math be a local coordinate system on the base manifold Template:Mvar with Template:Math and let

$\{\begin{array}{c}\psi :W\to \phi (U)\times {𝐑}^N\\ \psi \left(v^k{e}_k{|}_x\right):=(x^1,\dots ,x^n,v^1,\dots ,v^N)\end{array}$

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

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

so the fiber of the secondary vector bundle structure at Template:Mvar in Template:Math is of the form

$p_{*}^{-1}(X)=\{X^k\frac{\partial }{\partial x^k}{|}_v+Y^{\ell }\frac{\partial }{\partial v^{\ell }}{|}_v:v\in E_x;Y^1,\dots ,Y^N\in 𝐑\}.$

Now it turns out that

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

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

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

and

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

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

Linearity of connections on vector bundles

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

$\{\begin{array}{c}\kappa :T_{v}E\to E_{p(v)}\\ \kappa (X):={\mathrm{vl}}_v^{-1}(\mathrm{vpr}X)\end{array}$

where Template:Math is the vertical lift, and Template:Math is the vertical projection. The mapping

$\{\begin{array}{c}\mathrm{\nabla }:\mathrm{\Gamma }(TM)\times \mathrm{\Gamma }(E)\to \mathrm{\Gamma }(E)\\ {\mathrm{\nabla }}_{X}v:=\kappa (v_{*}X)\end{array}$

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

$\begin{array}{cc}{\mathrm{\nabla }}_{X+Y}v& ={\mathrm{\nabla }}_{X}v+{\mathrm{\nabla }}_{Y}v\\ {\mathrm{\nabla }}_{\lambda X}v& =\lambda {\mathrm{\nabla }}_{X}v\\ {\mathrm{\nabla }}_X(v+w)& ={\mathrm{\nabla }}_{X}v+{\mathrm{\nabla }}_{X}w\\ {\mathrm{\nabla }}_X(\lambda v)& =\lambda {\mathrm{\nabla }}_{X}v\\ {\mathrm{\nabla }}_X(fv)& =X[f]v+f{\mathrm{\nabla }}_{X}v\end{array}$

if and only if the connector map is linear with respect to the secondary vector bundle structure Template:Math on Template:Math. Then the connection is called linear. Note that the connector map is automatically linear with respect to the tangent bundle structure Template:Math.

See also

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

References

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