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.

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

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

of the original addition and scalar multiplication

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

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

$\psi :W\to \varphi (U)\times \mathbb {R} ^{N}\quad ;\quad \psi (v^{k}e_{k}|_{x}):=(x^{1},\ldots ,x^{n},v^{1},\ldots ,v^{N})$

be a coordinate system on E adapted to it. Then

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

so the fiber of the secondary vector bundle structure at X∈T_xM is of the form

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

Now it turns out that

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

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

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

and

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

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

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

where vl_v:E→V_vE is a natural vector space isomorphism

$\operatorname {vl} _{v}w[f]:={\frac {d}{dt}}{\Big |}_{t=0}f(v+tw)\quad ,\quad f\in C^{\infty }(E)$

known as the vertical lift, and vpr_v:T_vE→V_vE is the vertical projection. The mapping

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

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

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

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

• Double tangent bundle

References

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