Secondary vector bundle structure

Last updated February 21, 2019

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 $(TE, E, TM, M)$ .

Mathematics includes the study of such topics as quantity, structure, space, and change.

In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.

In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X : to every point x of the space X we associate a vector space V(x) in such a way that these vector spaces fit together to form another space of the same kind as X, which is then called a vector bundle over X.

In mathematics, particularly differential topology, the double tangent bundle or the second tangent bundle refers to the tangent bundle (TTM,π_TTM,TM) of the total space TM of the tangent bundle (TM,π_TM,M) of a smooth manifold M . A note on notation: in this article, we denote projection maps by their domains, e.g., π_TTM : TTM → TM. Some authors index these maps by their ranges instead, so for them, that map would be written π_TM.

In differential geometry, the tangent bundle of a differentiable manifold $is a manifold which assembles all the tangent vectors in M . As a set, it is given by the disjoint union of the tangent spaces of M . That is,$

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

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

\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

In differential geometry, an Ehresmann connection is a version of the notion of a connection, which makes sense on any smooth fiber bundle. In particular, it does not rely on the possible vector bundle structure of the underlying fiber bundle, but nevertheless, linear connections may be viewed as a special case. Another important special case of Ehresmann connections are principal connections on principal bundles, which are required to be equivariant in the principal Lie group action.

{\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

{\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

In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given by a principal connection on the frame bundle – see affine connection. In the special case of a manifold isometrically embedded into a higher-dimensional Euclidean space, the covariant derivative can be viewed as the orthogonal projection of the Euclidean derivative along a tangent vector onto the manifold's tangent space. In this case the Euclidean derivative is broken into two parts, the extrinsic normal component and the intrinsic covariant derivative component.

{\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 $(TE, π TE, E)$ .

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References

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

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