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

[math]\displaystyle{ +_*:T(E\times E)\to TE, \qquad \lambda_*:TE\to TE }[/math]

of the original addition and scalar multiplication

[math]\displaystyle{ +:E\times E\to E, \qquad \lambda:E\to E }[/math]

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

[math]\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} }[/math]

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

[math]\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)}, }[/math]

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

[math]\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 \}. }[/math]

Now it turns out that

[math]\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 ) }[/math]

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

[math]\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} }[/math]

and

[math]\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}, }[/math]

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

[math]\displaystyle{ \begin{cases}\kappa:T_vE\to E_{p(v)} \\ \kappa(X):=\operatorname{vl}_v^{-1}(\operatorname{vpr}X) \end{cases} }[/math]

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

[math]\displaystyle{ \begin{cases}\nabla:\Gamma(TM)\times\Gamma(E)\to\Gamma(E) \\ \nabla_Xv := \kappa(v_*X) \end{cases} }[/math]

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

[math]\displaystyle{ \begin{align} \nabla_{X+Y}v &= \nabla_X v + \nabla_Y v \\ \nabla_{\lambda X}v &=\lambda \nabla_Xv \\ \nabla_X(v+w) &= \nabla_X v + \nabla_X w \\ \nabla_X(\lambda v) &=\lambda \nabla_Xv \\ \nabla_X(fv) &= X[f]v + f\nabla_Xv \end{align} }[/math]

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).