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 $p_{*}:TE\to TM$ of the original projection map $p:E\to M$.

In the special case $(E,p,M)=(TM,\pi _{TM},M)$ where $TE=TTM$ is the double tangent bundle the secondary vector bundle $(TTM,(\pi _{TM})_{*},TM)$ is isomorphic to the tangent bundle $(TTM,\pi _{TTM},TM)$ 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 $p_{*}^{-1}(X)\subset TE$ of any tangent vector $X\in TM$ in the push-forward $p_{*}:TE\to TM$ of the canonical projection $p:E\to M$ 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 $(TE,p_{*},TN)$ becomes a smooth vector bundle with these vector space operations on its fibres.

Proof

Let $(U,\varphi )$ be a local coordinate system on the base manifold $M$ and with $\varphi (x)=(x^{1},\ldots ,x^{n})$ 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

$X=X^{k}{\frac {\partial }{\partial x^{k}}}{\Big |}_{x}\in T_{x}M$

is of the form

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

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 $\chi :TW\to TU\times \mathbb {R} ^{2N}$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

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

so each fibre $p_{*}^{-1}(X)\subset TE$ 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 the isomorphism $\operatorname {vl} _{v}:E\to V_{v}E$ is the vertical lift

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

and $\operatorname {vpr} _{v}:T_{v}E\to V_{v}E$ 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 $\Gamma (E)$ 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 $(TE,\pi _{TE},E)$.

See also

• Double tangent bundle

References

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