Samiksha Jaiswal (Editor)

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∗ : TE → TM of the original projection map p : E → M. This gives rise to a double vector bundle structure .

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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

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

+ ∗ : T ( E × E ) → T E , λ ∗ : T E → T E

of the original addition and scalar multiplication

+ : E × E → E , λ : E → 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) = (x1, ..., xn) and let

{ ψ : W → φ ( U ) × R N ψ ( v k e k | x ) := ( x 1 , … , x n , v 1 , … , v N )

be a coordinate system on W := p − 1 ( U ) ⊂ E adapted to it. Then

p ∗ ( X k ∂ ∂ x k | v + Y ℓ ∂ ∂ v ℓ | v ) = X k ∂ ∂ x k | p ( v ) ,

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

p ∗ − 1 ( X ) = { X k ∂ ∂ x k | v + Y ℓ ∂ ∂ v ℓ | v   :   v ∈ E x ; Y 1 , … , Y N ∈ R } .

Now it turns out that

χ ( X k ∂ ∂ x k | v + Y ℓ ∂ ∂ v ℓ | v ) = ( X k ∂ ∂ x k | p ( v ) , ( v 1 , … , v N , Y 1 , … , Y N ) )

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

( X k ∂ ∂ x k | v + Y ℓ ∂ ∂ v ℓ | v ) + ∗ ( X k ∂ ∂ x k | w + Z ℓ ∂ ∂ v ℓ | w ) = X k ∂ ∂ x k | v + w + ( Y ℓ + Z ℓ ) ∂ ∂ v ℓ | v + w

and

λ ∗ ( X k ∂ ∂ x k | v + Y ℓ ∂ ∂ v ℓ | v ) = X k ∂ ∂ x k | λ v + λ Y ℓ ∂ ∂ v ℓ | λ v ,

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

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

{ κ : T v E → E p ( v ) κ ( X ) := vl v − 1 ⁡ ( vpr ⁡ X )

where vlv : E → VvE is the vertical lift, and vprv : TvE → VvE is the vertical projection. The mapping

{ ∇ : T M × Γ ( E ) → Γ ( E ) ∇ X v := κ ( v ∗ X )

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

∇ X + Y v = ∇ X v + ∇ Y v ∇ λ X v = λ ∇ X v ∇ X ( v + w ) = ∇ X v + ∇ X w ∇ X ( λ v ) = λ ∇ X v ∇ X ( f v ) = X [ f ] v + f ∇ X v

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).

References

Secondary vector bundle structure Wikipedia
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