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 .

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

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

\begin{cases}\kappa:T_vE\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:TM\times\Gamma(E)\to\Gamma(E) \\ \nabla_Xv := \kappa(v_*X) \end{cases}

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

\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}

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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Secondary vector bundle structure

Construction of the secondary vector bundle structure

Proof

Linearity of connections on vector bundles

See also

References