Secondary vector bundle structure

Short description: Mathematical concept in particularly differential topology

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

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

of the original addition and scalar multiplication

$+:E{\times }_{M}E\to E,\lambda :E\to E$

as its vector space operations. It becomes clear ${+}_{*}$ actually defines addition on the fibers of $p_{*}$ as $T(E{\times }_{M}E)=TE{\times }_{TM}TE$. 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

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

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 X in T_xM is of the form

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

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 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

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

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 (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

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

where vl_v : E → V_vE is the vertical lift, and vpr_v : T_vE → V_vE is the vertical projection. The mapping

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

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

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

if and only if the connector map is linear with respect to the secondary vector bundle structure (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).

See also

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

References

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