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Secondary vector bundle structure

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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 : TETM of the original projection map p : EM. 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

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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 : TETM of the canonical projection p : EM is a smooth submanifold of dimension 2N, and it becomes a vector space with the push-forwards

of the original addition and scalar multiplication

as its vector space operations. The triple (TE, p, TM) becomes a smooth vector bundle with these vector space operations on its fibres.

Proof

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Let (U, φ) be a local coordinate system on the base manifold M with φ(x) = (x1, ..., xn) and let

be a coordinate system on adapted to it. Then

so the fiber of the secondary vector bundle structure at X in TxM is of the form

Now it turns out that

gives a local trivialization χ : TWTU × R2N for (TE, p, TM), and the push-forwards of the original vector space operations read in the adapted coordinates as

and

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

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The general Ehresmann connection TE = HEVE on a vector bundle (E, p, M) can be characterized in terms of the connector map

where vlv : EVvE is the vertical lift, and vprv : TvEVvE is the vertical projection. The mapping

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

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

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References

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