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 $p : E \to M$ . 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 $(p *) -1 (X) \subset TE$ 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 $2 N$ , and it becomes a vector space with the push-forwards

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

of the original addition and scalar multiplication

+: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 $φ (x) = (x 1, ..., x n)$ and let

{\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 $W:=p^{-1}(U)\subset E$ adapted to it. Then

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 $T x M$ 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}\

Now it turns out that

\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 $χ : TW \to TU \times R 2 N$ for $(TE, p *, TM)$ , and the push-forwards of the original vector space operations read in the adapted coordinates as

\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

\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 $(p *) -1 (X) \subset TE$ 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

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

where $vl v : E \to V v E$ is the vertical lift, and $vpr v : T v E \to V v E$ is the vertical projection. The mapping

{\displaystyle {\begin{cases}\nabla

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

{\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 $(TE, π TE, E)$ .

References

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

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Construction of the secondary vector bundle structure

Proof

Linearity of connections on vector bundles

See also

References