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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, πTM, M), where TE = TTM is the double tangent bundle, the secondary vector bundle (TTM, (πTM)∗, TM) is isomorphic to the tangent bundle (TTM, πTTM, TM) of TM through the canonical flip.

• Construction of the secondary vector bundle structure

• Proof

• Linearity of connections on vector bundles

• See also

• References

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

of the original addition and scalar multiplication

+ : E × × M E → → E , λ λ : E → → E {\displaystyle +:E\times _{\!M}\!E\to E,\qquad \lambda :E\to E}

as its vector space operations. It becomes clear + ∗ ∗ {\displaystyle +_{*}} actually defines addition on the fibers of p ∗ ∗ {\displaystyle p_{*}} as T ( E × × M E ) = T E × × T M T E {\displaystyle 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.

Let (U, φ) be a local coordinate system on the base manifold M with φ(x) = (x1, ..., xn) and let

{ ψ ψ : W → → φ φ ( U ) × × R N ψ 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 ℓ ∂ ∂ ∂ ∂ v ℓ 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 ℓ ∂ ∂ ∂ ∂ v ℓ 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 χ ( X k ∂ ∂ ∂ ∂ x k | v + Y ℓ ℓ ∂ ∂ ∂ ∂ v ℓ 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 ℓ ∂ ∂ ∂ ∂ v ℓ ℓ | v ) + ∗ ∗ ( X k ∂ ∂ ∂ ∂ x k | w + Z ℓ ℓ ∂ ∂ ∂ ∂ v ℓ ℓ | w ) = X k ∂ ∂ ∂ ∂ x k | v + w + ( Y ℓ ℓ + Z ℓ ℓ ) ∂ ∂ ∂ ∂ v ℓ 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 λ ∗ ∗ ( X k ∂ ∂ ∂ ∂ x k | v + Y ℓ ℓ ∂ ∂ ∂ ∂ v ℓ ℓ | v ) = X k ∂ ∂ ∂ ∂ x k | λ λ v + λ λ Y ℓ ℓ ∂ ∂ ∂ ∂ v ℓ ℓ | λ 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.

The general Ehresmann connection TE = HE ⊕ VE on a vector bundle (E, p, M) can be characterized in terms of the connector map

{ κ κ : T v E → → E p ( v ) κ 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 Γ ( T M ) × × Γ Γ ( E ) → → Γ Γ ( E ) ∇ ∇ X v := κ 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 λ X v = λ λ ∇ ∇ X v ∇ ∇ X ( v + w ) = ∇ ∇ X v + ∇ ∇ X w ∇ ∇ X ( λ λ v ) = λ 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).

• Connection (vector bundle)

• Double tangent bundle

• Ehresmann connection

• Vector bundle

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