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. 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

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

of the original addition and scalar multiplication

${\displaystyle +:E\times E\to E,\qquad \lambda :E\to E}$

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

${\displaystyle {\begin{cases}\psi :W\to \varphi (U)\times \mathbf {R} ^{N}\\\psi \left(v^{k}e_{k}|_{x}\right):=\left(x^{1},\ldots ,x^{n},v^{1},\ldots ,v^{N}\right)\end{cases}}}$

be a coordinate system on ${\displaystyle W:=p^{-1}(U)\subset E}$ adapted to it. Then

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

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

${\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

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

gives a local trivialization χ : TW → TU × R^2N for (TE, p_∗, TM), and the push-forwards of the original vector space operations read in the adapted coordinates as

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

and

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

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

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

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

${\displaystyle {\begin{cases}\nabla$ :\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

${\displaystyle {\begin{aligned}\nabla _{X+Y}v&=\nabla _{X}v+\nabla _{Y}v\\\nabla _{\lambda X}v&=\lambda \nabla _{X}v\\\nabla _{X}(v+w)&=\nabla _{X}v+\nabla _{X}w\\\nabla _{X}(\lambda v)&=\lambda \nabla _{X}v\\\nabla _{X}(fv)&=X[f]v+f\nabla _{X}v\end{aligned}}}$

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, 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).