Harmonious set

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.

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∈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

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

of the original addition and scalar multiplication

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

as its vector space operations. The triple (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

$\psi :W\to \phi (U)\times {\mathbb{ℝ}}^N;\psi (v^k{e}_k{|}_x):=(x^1,\dots ,x^n,v^1,\dots ,v^N)$

be a coordinate system on 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∈T_xM is of the form

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

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

$TE=HE\oplus VE$

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

$\kappa :T_{v}E\to E_{p(v)};\kappa (X):={\mathrm{vl}}_v^{-1}(\mathrm{vpr}X),$

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

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

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

• ${\mathrm{\nabla }}_{X+Y}v={\mathrm{\nabla }}_{X}v+{\mathrm{\nabla }}_{Y}v$
• ${\mathrm{\nabla }}_{\lambda X}v=\lambda {\mathrm{\nabla }}_{X}v$
• ${\mathrm{\nabla }}_X(v+w)={\mathrm{\nabla }}_{X}v+{\mathrm{\nabla }}_{X}w$
• ${\mathrm{\nabla }}_X(\lambda v)=\lambda {\mathrm{\nabla }}_{X}v$
• ${\mathrm{\nabla }}_X(fv)=X[f]v+f{\mathrm{\nabla }}_{X}v$

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).