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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 × E ) → T E , λ ∗ : T E → T E {\displaystyle +_{*}:T(E\times E)\to TE,\qquad \lambda _{*}:TE\to TE}

of the original addition and scalar multiplication

+ : E × E → E , λ : E → E {\displaystyle +: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) = (x1, ..., xn) and let

{ ψ : W → φ ( U ) × R N ψ ( v k e k | x ) := ( x 1 , … , x n , v 1 , … , v N ) {\displaystyle {\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 ) ⊂ E {\displaystyle W:=p^{-1}(U)\subset E} adapted to it. Then

p ∗ ( X k ∂ ∂ x k | v + Y ℓ ∂ ∂ v ℓ | v ) = X k ∂ ∂ x k | p ( v ) , {\displaystyle 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 TxM is of the form

p ∗ − 1 ( X ) = { X k ∂ ∂ x k | v + Y ℓ ∂ ∂ v ℓ | v   :   v ∈ E x ; Y 1 , … , Y N ∈ R } . {\displaystyle 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

χ ( X k ∂ ∂ x k | v + Y ℓ ∂ ∂ v ℓ | v ) = ( X k ∂ ∂ x k | p ( v ) , ( v 1 , … , v N , Y 1 , … , Y N ) ) {\displaystyle \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 × R2N for (TE, p∗, TM), and the push-forwards of the original vector space operations read in the adapted coordinates as

( X k ∂ ∂ x k | v + Y ℓ ∂ ∂ v ℓ | v ) + ∗ ( X k ∂ ∂ x k | w + Z ℓ ∂ ∂ v ℓ | w ) = X k ∂ ∂ x k | v + w + ( Y ℓ + Z ℓ ) ∂ ∂ v ℓ | v + w {\displaystyle \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

λ ∗ ( X k ∂ ∂ x k | v + Y ℓ ∂ ∂ v ℓ | v ) = X k ∂ ∂ x k | λ v + λ Y ℓ ∂ ∂ v ℓ | λ v , {\displaystyle \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

{ κ : T v E → E p ( v ) κ ( X ) := vl v − 1 ⁡ ( vpr ⁡ X ) {\displaystyle {\begin{cases}\kappa :T_{v}E\to E_{p(v)}\\\kappa (X):=\operatorname {vl} _{v}^{-1}(\operatorname {vpr} X)\end{cases}}}

where vlv : E → VvE is the vertical lift, and vprv : TvE → VvE is the vertical projection. The mapping

{ ∇ : Γ ( T M ) × Γ ( E ) → Γ ( E ) ∇ X v := κ ( v ∗ X ) {\displaystyle {\begin{cases}\nabla :\Gamma (TM)\times \Gamma (E)\to \Gamma (E)\\\nabla _{X}v:=\kappa (v_{*}X)\end{cases}}}

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

∇ X + Y v = ∇ X v + ∇ Y v ∇ λ X v = λ ∇ X v ∇ X ( v + w ) = ∇ X v + ∇ X w ∇ X ( λ v ) = λ ∇ X v ∇ X ( f v ) = X [ f ] v + f ∇ X v {\displaystyle {\begin{aligned}\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\end{aligned}}}

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, πTE, E).

See also

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

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