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[edit]
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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e84760bcd5520c355a2e94407e1a080e1f7259b0)
of the original addition and scalar multiplication
![{\displaystyle +:E\times E\to E,\qquad \lambda :E\to E}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5629aec450c27329949540ed5de60209b2f99491)
as its vector space operations. 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
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3cf5b2525f068e3c3c40fa9af14d3d3215b7fce2)
be a coordinate system on
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)},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d467a3150cd4711c6e971fe5c96c5c9e18bd5a7)
so the fiber of the secondary vector bundle structure at X in TxM 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\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/084216fc7964d1eac65b6cb9dad0b21e865f0f50)
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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/21e8bbae94a9be89e92c89a7bf01a27b12306098)
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
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bab554a067ea624a2f8ba88c95bb5b9843e96cd2)
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},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd017cb6d6b4e4f19a7f9a7221069263a28f9b12)
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[edit]
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab1e2913d4b4496de961cf47e6001214c5de429a)
where vlv : E → VvE is the vertical lift, and vprv : TvE → VvE 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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f8589773e4471b963b93f703789eaacd1d475669)
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f781e04dac95200da9a61a5a42954c9e42bf364)
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).
See also[edit]
References[edit]
- P.Michor. Topics in Differential Geometry, American Mathematical Society (2008).